Finite Elements

Guido Kanschat

May 7, 2019 Contents

1 Elliptic PDE and Their Weak Formulation 3 1.1 Elliptic boundary value problems ...... 3 1.1.1 Linear second order PDE ...... 3 1.1.2 Variational principle and weak formulation ...... 8 1.1.3 Boundary conditions in weak form ...... 10 1.2 Hilbert Spaces and Bilinear Forms ...... 11 1.3 Fast Facts on Sobolev Spaces ...... 20 1.4 Regularity of Weak Solutions ...... 24

2 Conforming Finite Element Methods 27 2.1 Meshes, shape functions, and degrees of freedom ...... 27 2.1.1 Shape function spaces on simplices ...... 30 2.1.2 Shape functions on tensor product cells ...... 31 2.1.3 The Galerkin equations and Céa’s lemma ...... 35 2.1.4 Mapped ﬁnite elements ...... 37 2.2 A priori error analysis ...... 40 2.2.1 Approximation of Sobolev spaces by ﬁnite elements . . . . 41 2.2.2 Estimates of stronger norms ...... 45 2.2.3 Estimates of weaker norms and linear functionals . . . . . 46 2.2.4 Green’s function and maximum norm estimates ...... 48 2.3 A posteriori error analysis ...... 50 2.3.1 Quasi-interpolation in H1 ...... 50

1 3 Variational Crimes 59

3.1 Numerical quadrature ...... 59

4 Solving the Discrete Problem 67

4.1 The Richardson iteration ...... 70

4.2 The conjugate gradient method ...... 75

4.3 Condition numbers of ﬁnite element matrices ...... 79

4.4 Multigrid methods ...... 82

5 Discontinuous Galerkin methods 84

5.1 Nitsche’s method ...... 84

5.2 The interior penalty method ...... 89

5.2.1 Bounded formulation in H1 ...... 93

2 Chapter 1

Elliptic PDE and Their Weak Formulation

1.1 Elliptic boundary value problems

1.1.1 Linear second order PDE

1.1.1 Notation: Dimension of “physical space” will be denoted by d. We denote coordinates in Rd as

T x = (x1, . . . , xd) .

In the special cases d = 2, 3 we also write

x x x = , x = y , y   z

respectively. The Euclidean norm on Rd is denoted as v u d uX 2 |x| = t xi . i=1

3 1.1.2 Notation: Partial derivatives of a function u ∈ C1(Rd) are denoted by

∂u(x) ∂ = ∂x u(x) = ∂xi u(x) = ∂iu(x). ∂xi i The gradient of u ∈ C1 is the row vector

∇u = (∂1u, . . . , ∂du)

The Laplacian of a function u ∈ C2(Rd) is

d 2 2 X 2 ∆u = ∂1 u + ··· + ∂du = ∂i u i=1

1.1.3 Notation: When we write equations, we typically omit the independent variable x. Therefore,

∆u ≡ ∆u(x).

1.1.4 Deﬁnition: A linear PDE of second order in divergence form for a function u ∈ C2(Rd) is an equation of the form

d d X X − ∂i aij(x)∂ju + bi(x)∂iu + c(x)u = f(x) (1.1) i,j=1 i=1

1.1.5 Deﬁnition: An important model problem for the equations we are going to study is Poisson’s equation

−∆u = f. (1.2)

1.1.6. Already with ordinary diﬀerential equations we experience that we typically do not search for solutions of the equation itself, but that we “anchor” the solution by solving an initial value problem, ﬁxing the solution at one point on the time axis.

It does not make sense to speak about an initial point in Rd. Instead, it turns out that it is appropriate to consider solutions on certain subsets of Rd and impose conditions at the boundary.

4 1.1.7 Deﬁnition: A domain in Rd is a connected, open set of Rd. We typically use the notation Ω ⊂ Rd. The boundary of a domain Ω is denoted by ∂Ω. To any point x ∈ ∂Ω, we associate the outer unit normal vector n ≡ n(x). The symbol ∂nu ≡ (∇u)n denotes the normal derivative of a function u ∈ C1(Ω) at a point x ∈ ∂Ω.

1.1.8 Deﬁnition: We distinguish three types of boundary conditions for Poisson’s equation, namely for a point x ∈ ∂Ω with a given function g 1. Dirichlet:

u(x) = g(x)

2. Neumann:

∂nu(x) = g(x)

3. Robin: for some positive function α on dΩ

∂nu(x) + α(x)u(x) = g(x)

While only one of these boundary conditions can hold in a single point x, diﬀerent boundary conditions can be active on diﬀerent subsets of ∂Ω. We denote such subsets as ΓD, ΓN , and ΓR.

1.1.9 Definition: The Dirichlet problem for Poisson’s equation (in differential form) is: find u ∈ C2(Ω) ∩ C(Ω), such that

−∆u(x) = f(x) x ∈ Ω, (1.3a) u(x) = g(x) x ∈ ∂Ω. (1.3b)

Here, the functions f on Ω and g on ∂Ω are data of the problem. The Dirichlet problem is called homogeneous, if g ≡ 0.

5 1.1.10 Theorem (Dirichlet principle): If a function u ∈ C2(Ω) ∩ C(Ω) solves the Dirichlet problem, then it minimizes the Dirichlet energy Z Z 1 2 E(v) = 2 |∇v| dx − fv dx, (1.4) Ω Ω among all functions v from the set

2 Vg = v ∈ C (Ω) ∩ C(Ω) v|∂Ω = g . (1.5)

This minimizer is unique.

Proof. Using variation of E, we will show that d E(u + εv) = 0 dε ε=0 for all v ∈ V0 since this implies u + εv = g on ∂Ω. By evaluating the square we have d Z E(u + εv) = ∇u∇v + ε|∇v|2 − fv dx dε Ω Since we are intersted in E(u), we now consider ε = 0. We get that u minimizes E(u + εv) at ε = 0 implies Z Z ∇u∇v dx = fv dx, ∀v ∈ V0. Ω Ω By Green’s formula Z Z Z ∇u∇v dx = −∆uv dx + ∂nuv ds Ω Ω ∂Ω we obtain that if u minimizes E(·), then Z Z ∇u∇v dx = fv dx, ∀v ∈ V0, Ω Ω since v ∈ V0 vanishes on ∂Ω. In summary, we have proven so far that if u solves Poisson’s Equation, then it is a stationary point of E(·). It remains to show R R that E(u) ≤ E(u + v) for any v ∈ V0. Using Ω fv dx = Ω ∇u∇v yields

1 Z 1 E(u+v)−E(u) = |∇(u+v)|2−2∇(u+v)∇u+|∇u|2 dx = |∇v|2 dx ≥ 0. 2 Ω 2 This also proves uniqueness.

6 1.1.11 Lemma: A minimizing sequence for the Dirichlet energy exists and it is a Cauchy sequence.

Proof. The Dirichlet energy E(·) is bounded from below and hence an inﬁnum (n) exists. Thus, there also exists a series {u }n∈N converging to this inﬁnum, i.e. lim E(u(n)) = inf E(v). n→∞ v∈V0

(n) Second, we show that {u }n is a Cauchy sequence. For the ﬁrst part we use Friedrich’s inequality

kvkL2(Ω) ≤ λ(Ω)k∇vkL2(Ω) v ∈ V0. The proof of this result will be given later. Using Hölder’s inequality we obtain Z 1 2 1 2 E(v) = k∇vkL2(Ω − fv dx ≥ k∇vkL2(Ω − kfkL2(ΩkvkL2(Ω 2 Ω 2 Applying Friedrich’s inquality yields that the above expression is greater or equal than

1 2 1 k∇vk 2 − k∇vk 2 kfk 2 . 2 L (Ω L (Ω λ(Ω) L (Ω

2 2 Finally, we apply Young’s inequality ab ≤ 1/2(a + b ) to obtain

1 2 2 1 2 k∇vk 2 − k∇vk 2 − kfk 2 2 L (Ω L (Ω 2λ(Ω) L (Ω

1 2 which yields E(v) ≥ − 2λ(Ω)2 kfkL2(Ω as a lower bound independent of v. To prove the second part, we use the parallelogram identity |v + w|2 + |v − w|2 = 2|v|2 + 2|w|2. Let m, n be natural numbers, then

(n) (m) 2 (n) 2 (m) 2 1 (n) (m) 2 |u − u |1 =2|u |1 + 2|u |1 − 4| /2(u + u |1 Z Z =4E(u(n)) + 4 fu(n) dx + 4E(u(m)) + 4 fu(m) dx Z (n) (m) (n) (m) − 8E(1/2(u + u ) − 8 1/2f(u + u )

(n) (m) (n) (m) =4E(u ) + 4E(u ) − 8E(1/2f(u + u ))

(n) (m) Taking the limit m, n → ∞ yields 4E(u ) + 4E(u ) → 8 infv∈V0 E(v). 1 (n) (m) Lastly, −E( /2f(u + u )) can be bounded by infv∈V0 E(v). It follows that (n) (m) 2 lim supm,n→∞|u − u |1 ≤ 0 and consequently as desired

(n) (m) 2 lim |u − u |1 = 0. m,n→∞

7 Note 1.1.12. Dirichlet-proof Dirichlet’s principle proved essential for the de- velopment of a rigorous solution theory for Poisson’s equation. Its proof will be deferred to the next theorem.

1.1.2 Variational principle and weak formulation

1.1.13 Theorem: A function u ∈ Vg minimizes the Dirichlet energy, if and only if there holds Z Z ∇u · ∇v dx = fv dx, ∀v ∈ V0. (1.6) Ω Ω Moreover, any solution to the Dirichlet problem in Deﬁnition 1.1.9 solves this equation.

1.1.14 Corollary: If a minimizer of the Dirichlet energy exists, it is necessarily unique.

1.1.15 Lemma: A function u ∈ Vg minimizes the Dirichlet energy admits the representation u = ug + u0, where ug ∈ Vg is arbitrary and u0 ∈ V0 solves Z Z Z ∇u · ∇v dx = fv dx − ∇ug · ∇v dx, ∀v ∈ V0. (1.7) Ω Ω Ω

The function u0 depends on the choice of ug, but not the minimizer u.

1.1.16 Notation: The inner product of L2(Ω) is denoted by Z (u, v) ≡ (u, v)Ω ≡ (u, v)L2(Ω) = uv dx. Ω Its norm is q kuk ≡ kukΩ ≡ kukL2(Ω) ≡ kukL2 = (u, v)L2(Ω).

1.1.17 Lemma (Friedrichs inequality): For any function in v ∈ V0 there holds

kvkΩ ≤ diam(Ω)k∇vkΩ. (1.8)

8 1.1.18 Lemma: The deﬁnitions

|v|1 = k∇vkL2(Ω), (1.9) q 2 2 kvk1 = kvkL2(Ω) + |v|1,

both deﬁne a norm on V0.

1.1.19 Problem: Prove the Friedrichs inequality.

1.1.20 Lemma: The Dirichlet energy with homogeneous boundary conditions is bounded from below and thus has an inﬁmum. In particular, there exists a minimizing sequence {un} such that as n → ∞,

E(un) → inf E(v). (1.10) v∈V0

1.1.21 Lemma: The minimizing sequence for the Dirichlet energy is a Cauchy sequence.

1.1.22 Deﬁnition: The completion of V0 under the norm kvk1 is the 1 Sobolev space H0 (Ω).

1 1.1.23 Lemma (Friedrichs inequality): For any function in v ∈ H0 there holds

kvkΩ ≤ diam(Ω)k∇vkΩ. (1.11)

1 1 Proof. Let v ∈ H0 (Ω). We make use of the fact, that by deﬁnition of H0 (Ω), there is a sequence vn → v with vn ∈ V0. By Lemma 1.1.17, Friedrichs’ inequality holds for vn uniformly in n. We conclude

kvkΩ ≤ kv − vnkΩ + kvnkΩ

≤ kv − vnkΩ + diam Ωk∇vnkΩ ≤ kv − vnkΩ + diam Ω k∇vn − ∇vkΩ + k∇vkΩ

As n → ∞, the norms of the diﬀerences converge to zero, such that the desired result holds in the limit.

9 1.1.24 Deﬁnition: The Dirichlet problem for Poisson’s equation in 1 weak form reads: ﬁnd u ∈ Hg (Ω) such that Z Z 1 ∇u · ∇v dx = fv dx, ∀v ∈ H0 (Ω). (1.12) Ω Ω

1.1.25 Theorem: The weak formulation in Deﬁnition 1.1.24 has a unique solution.

1.1.3 Boundary conditions in weak form

1.1.26 Lemma: Let u ∈ V = H1(Ω) be a solution to the weak formulation Z Z ∇u · ∇v dx = fv dx, ∀v ∈ V (Ω). (1.13) Ω Ω

If u ∈ C2(Ω)∩C1(Ω) and Ω has C1-boundary, then u solves the boundary value problem

−∆u = f in Ω (1.14) ∂nu = 0 on ∂Ω.

1.1.27 Deﬁnition: A boundary condition inherent in the weak formulation and not explicitly stated is called natural boundary condition. If boundary values are obtained by constraining the function space it is called essential boundary condition. We also call a boundary condition in strong form, if it is a constraint on the function space, and in weak form, if it is part of the weak formulation.

Remark 1.1.28. Dirichlet and homogeneous Neumann boundary conditions are examples for essential and natural boundary conditions, respectively.

10 1.1.29 Lemma: The boundary value problem

−∆u = f in Ω

u = 0 on ΓD ⊂ ∂Ω (1.15)

∂nu + αu = g on ΓR ⊂ ∂Ω,

has the weak form: ﬁnd u ∈ V such that Z Z Z Z ∇u · ∇v dx + αuv ds = fv dx + gv ds, ∀v ∈ V (Ω). Ω ΓR Ω ΓR (1.16)

1.2 Hilbert Spaces and Bilinear Forms

1.2.1 Deﬁnition: Let V be a vector space over R. An inner product on V is a mapping h., .i : V × V → R with the properties

hαx + y, zi = αhx, zi + hy, zi ∀x, y, z ∈ V ; α ∈ K (1.17) hx, yi = hy, xi ∀x, y ∈ V (1.18) hx, xi ≥ 0 ∀x ∈ V and (1.19) hx, xi = 0 ⇔ x = 0, (1.20)

usually referred to as (bi-)linearity, symmetry, and positive deﬁniteness. We note that linearity in the second argument follows immediately by symmetry.

1.2.2 Theorem (Bunyakovsky-Cauchy-Schwarz inequality): For every inner product there holds the inequality

hv, wi ≤ phv, viphw, wi. (1.21)

Equality holds if and only if v and w are collinear.

Proof. The case w = 0 is trivial. Without loss of generality we can therefore hv,wi assume that w 6= 0. Deﬁne λ ∈ R as λ = hw,wi . By (1.19) we have

0 ≤ hv − λw, v − λwi

11 and by (1.18) the right-hand side extends to

hv, vi − hv, λwi − hλw, vi + hλw, λwi = hv, vi − λhv, wi − λhv, wi + λλhw, wi.

Evaluating λ yields the inequality

hv, wihv, wi hv, wihv, wi hv, wihv, wihw, wi 0 ≤ hv, vi − − + . hw, wi hw, wi hw, wi2

The result follows from multiplication with hw, wi and arranging the summands.

For the second part let v, w be colinear, i.e. there is a λ ∈ K such that v = λw. Then deducing the equality is trivial. Now let equality hold for (1.21). We immediately get that the equality must also hold for

0 = hv − λw, v − λwi.

However, by (1.20) this implies

0 = v − λw.

Thus, v and w are colinear.

1.2.3 Lemma: Every inner product deﬁnes a norm by

kvk = phv, vi. (1.22)

Proof. Deﬁniteness and homogeneity follow from the properties of the inner product. It remains to show the triangle inequality

ku + vk ≤ kuk + kvk.

Squaring the left hand side yields with the Bunyakovsky-Cauchy-Schwarz inequality

ku + vk2 = hu + v, u + vi = kuk2 + 2hu, vi + kvk2 2 ≤ kuk2 + 2kukkvk + kvk2 = kuk + kvk .

12 1.2.4 Deﬁnition: A space V with is complete with respect to a norm, if all Cauchy sequences with elements in V have their limit in V .A subspace W ⊂ V is closed if it is complete in the topology of V . The completion of a space V with respect to a norm consists of the space V and the limits of all Cauchy sequences in V . We denote the completion of a space V by

k·k V = V V . (1.23)

1.2.5 Deﬁnition: A normed vector space is a vector space V with a norm k·k. We may also write k·kV to highlight the connection. A normed vector space V which is complete with respect to its norm is called a Banach space. A vector space V equipped with an inner product h., .i is called an inner product space or pre-Hilbert space.A Hilbert space is a pre- Hilbert space which is also complete.

1.2.6 Definition: Let V be an inner product space over a field K. Two vectors x, y ∈ V are called orthogonal if hx, yi = 0. We write x ⊥ y. Let W be a subspace of V . We say that a vector v is orthogonal to the subspace W , if it is orthogonal to every vector in W . A set of nonzero mutually orthogonal vectors {xi} ⊂ V is called orthogonal set. If additionally kxik = 1 for all vectors, it is called an orthonormal set. These notions transfer directly from finite to count- able sets.

1.2.7 Deﬁnition: Let W ⊂ V be a subspace of a Hilbert space V . We deﬁne its orthogonal complement W ⊥ ⊂ V by

⊥ W = v ∈ V hv, wiV = 0 ∀ w ∈ W . (1.24)

1.2.8 Lemma: The orthogonal complement W ⊥ of a subspace W ⊂ V is closed in the sense of Deﬁnition 1.2.4.

Proof. By the Bunyakovsky-Cauchy-Schwarz inequality, the inner product is continuous on V × V . Therefore, the mapping

ϕw : V → R, v 7→ hv, wi,

13 is continuous. For any w ∈ W , the kernel of ϕw is closed as the pre-image of the closed set {0}. Since

⊥ \ W = ker (ϕw) , w∈W it is closed as the intersection of closed sets.

1.2.9 Theorem: Let W be a subspace of a Hilbert space V and W ⊥ ⊥ its orthogonal complement. Then, W ⊥ = W . Further, V = W ⊕ W ⊥ if and only if W is closed.

⊥ Proof. Clearly, W ⊂ W ⊥ since W ⊂ W . Let now u ∈ W ⊥. Then, ϕ = hu, ·i is a continuous linear functional on V . Therefore, if a sequence wn ⊂ W converges to w ∈ W , we have

hu, wi = lim hu, wni = 0, n→∞

⊥ ⊥ since u ∈ W ⊥. Hence, u ∈ W and W ⊥ = W . Now, the “only if” follows by the fact, that if W is not closed, there is an element w ∈ W but not in W such that hw, ui = 0 for all u ∈ W ⊥. Thus, w 6∈ W ⊥ and consequently w 6∈ W ⊥ ⊕ W . Let now W be closed. We show that for all v ∈ V there is a unique decomposition

v = w + u, with w ∈ W, u ∈ W ⊥. (1.25)

This is equivalent to V = W ⊕ W ⊥. Uniqueness follows, since

v = w1 + u1 = w2 + u2 implies that for any y ∈ V

0 = hw1 − w2 + u1 − u2, yi = hw1 − w2, yi + hu1 − u2, yi.

Choosing y = u1 −u2 and w1 −w2 in turns, we see that one of the inner products vanishes for orthogonality and the other implies that the diﬀerence is zero. If v ∈ W , we choose w = v and u = 0. For v 6∈ W , we prove existence by considering that due to the closedness of W there holds

d = inf kv − w0k > 0. w0∈W

Let wn be a minimizing sequence. Using the parallelogram identity ka + bk2 + ka − bk2 = 2kak2 + 2kbk2,

14 we prove that {wn} is a Cauchy sequence by

2 2 kwm − wnk = k(v − wn) − (v − wm)k 2 2 2 = 2kv − wnk + 2kv − wmk − k2v − wm − wnk 2 2 2 wm + wn = 2kv − wnk + 2kv − wmk − 4 v − 2 2 2 2 ≤ 2kv − wnk + 2kv − wmk − 4d , since (wm + wn)/2 ∈ W and d is the inﬁmum. Now we use the minimizing property to obtain

2 2 2 2 lim kwm − wnk = 2d + 2d − 4d = 0. m,n→∞

Since V is given as a Hilbert space and as such complete, w = lim wn exists and by the closedness of W , we have w ∈ W . Let u = v − w. By continuity of the norm, we have kuk = d. It remains to show that u ∈ W ⊥. To this end, we introduce the variation w + εw˜ with w˜ ∈ W to obtain

d2 ≤ kv − w − εw˜k2 = kuk2 − 2εhu, w˜i + ε2kw˜k, implying for any ε > 0

0 ≤ −2εhu, w˜i + ε2kw˜k, which requires hu, w˜i = 0. Since w˜ ∈ W was chosen arbitrarily, we have u ∈ W ⊥.

1.2.10 Deﬁnition: Let W be a closed subspace of the Hilbert space V and W ⊥ be its orthogonal complement. Then, the orthogonal projection operators

ΠW : V → W ⊥ (1.26) ΠW ⊥ : V → W

are deﬁned by the unique decomposition

v = ΠW v + ΠW ⊥ v. (1.27)

15 1.2.11 Deﬁnition: A linear functional on a vector space V is a linear mapping from V to K. The dual space V ∗ of a vector space V , also called the normed dual, is the space of all bounded linear functionals on V equipped with the norm ϕ(v) kϕkV ∗ = sup . (1.28) v∈V kvkV

1.2.12 Theorem (Riesz representation theorem): Let V be a Hilbert space. Then, V is isometrically isomorphic to V ∗. In particular, there is an isomorphism

%: V → V ∗, (1.29) y 7→ f,

such that hx, yi = f(x) ∀x ∈ V, (1.30) kykV = kfkV ∗ .

We refer to % as Riesz isomorphism.

Proof. The proof is constructive and makes use of the orthogonal complement.

First, it is clear that for any y ∈ V a linear functional f ∈ V ∗ is deﬁned by f(·) = h·, yi. Furthermore, % is injective, since

hx, yi = 0 ∀x ∈ V implies y ∈ V ⊥ = {0}. By the Bunyakovsky-Cauchy-Schwarz inequality, we have |f(x)| |hx, yi| kfkV ∗ = sup = sup ≤ kykV , x∈V kxkV x∈V kxkV with equality for x = y. It remains to show that % is surjective. To this end, let f ∈ V ∗ be arbitrary and let N = ker (f). If N = V , we choose y = 0. If not, choose y⊥ ∈ N ⊥ and let

f(y⊥) y = y⊥ ∈ N ⊥, (1.31) ky⊥k2

16 2 2 such that f(y) = f(y⊥) / y⊥ 6= 0. Let now x ∈ V be chosen arbitrarily. Then, there holds

f(x) f(x) x = x − y + y, f(y) f(y)

f(x) where f(y) denotes a scalar. Since

f(x) f(y) f x − y = f(x) − f(x) = 0, f(y) f(y) this decomposition amounts to x = x0 + x⊥ with x0 ∈ N and x⊥ ∈ N ⊥. It is unique according to Deﬁnition 1.2.10. Thus, we have that x⊥ is a multiple of ⊥ f(x) y, say x = αy with α = f(y) and thus

2 f(y⊥) f(x) = f(x0) + f(x⊥) =αf(y) =α ky⊥k2 ⊥ 2 0 ⊥ 2 f(y ) ⊥ 2 hx, yi = x , y + x , y =αkykV =α y ky⊥k4

Hence, the two terms are equal and % is surjective.

1.2.13 Deﬁnition: A bilinear form a(., .) on a Hilbert space V is a mapping a: V × V → R, which is linear in both arguments. The bilinear form is bounded, if there is a constant M such that

a(u, v) ≤ MkukV kvkV , ∀u, v ∈ V. (1.32)

It is called coercive or elliptic, if there is a constant α such that

2 a(u, u) ≥ αkukV ∀u ∈ V. (1.33)

Notation 1.2.14. One often speaks of V -elliptic instead of elliptic to point out that the ellipticity of the bilinear form indeed depends on the scalarproduct inducing the V -norm.

17 1 0 1.2.15 Lemma: Let aij ∈ C (Ω), bi, c ∈ C (Ω). A solution to the Dirichlet problem

d d X X − ∂i aij∂ju + bi∂iu + cu = f in Ω i,j=1 i=1 (1.34) u = 0 on ∂Ω

solves the weak problem: ﬁnd u ∈ V0 such that for all v ∈ V0

a(u, v) ≡ (A∇u, ∇v) + (b · ∇u, v) + (cu, v) = (f, v), (1.35) where A(x) = aij(x) is the matrix of coefficients of the second order term and b(x) = bi(x) is the vector of coefficients of the first order term. If additionally u ∈ C2(Ω) holds, then the solution to the weak problem solves the Dirichlet problem in differential form.

1.2.16 Lemma (Lax-Milgram): Let a(., .) be a bounded, coercive bilinear form on a Hilbert space V and let f ∈ V ∗. Then, there is a unique element u ∈ V such that

a(u, v) = f(v) ∀v ∈ V. (1.36)

Furthermore, there holds 1 kuk ≤ kfk ∗ . (1.37) V α V

Proof. To prove Lax-Milgram we ﬁrst consider uniqueness and then the existence of a solution.

Assume that there are solutions u1, u2 ∈ V of (1.36), i. e. there holds a(u1, v) = f(v) and a(u2, v) = f(v) for all v ∈ V . Thus, a(u1 − u2, v) = 0 for all v ∈ V . Now choose v = u1 − u2 ∈ V . Since a(., .) is coercive with α > 0 it holds

2 0 = a(u1 − u2, u1 − u2) ≥ αku1 − u2kV which implies u1 − u2 = 0. Hence u1 = u2. Let us now consider the existence of a solution. We will deﬁne a linear functional to apply Riesz and Banach ﬁxed point theorem. For all y ∈ V there holds

hy, ·i − ω [a(y, ·) − f(·)] ∈ V ∗

18 with ω > 0. Due to Riesz there exists an isomorphism % : V → V ∗ that maps a given z ∈ V to hy, ·i − ω [a(y, ·) − f(·)] such that hv, zi = hy, vi − ω [a(y, v) − f(v)] ∀v ∈ V.

Now we deﬁne the mapping Tω : V → V that maps y 7→ z and deﬁne S : V → V such that hSu, vi = a(u, v) ∀v ∈ V with kSukV ≤ MkukV for M > 0. Now consider y − x instead of y. This leads for all v ∈ V to

hTωy − Tωx, vi = hy − x, vi − ω [a(y − x, v)] = hy − x, vi − ωhS(y − x), vi.

Thus, we can conclude that Tω(y − x) = y − x − ωS(y − x). Applying the norm and using the fact that it is induced by the inner product of V we get 2 2 2 2 kTωy − TωxkV = ky − xkV − 2ωhS(y − x), y − xi + ω kS(y − x)k ≤ ky − xk2 − 2ωa(y − x, y − x) + ω2M 2ky − xk2 ≤ (1 − 2ωα + ω2M 2)ky − xk2.

As we want to apply Banach ﬁxed point theorem, we need Tω to be a contraction. 2 2 2α Therefore, we need 1 − 2ωα + ω M < 1 which is given for ω ∈ 0, M 2 . Hence, Tω is a contraction and there exists a ν ∈ V such that Tων = ν and hTων, νi = hν, νi − ω [a(ν, v) − f(v)] which implies a(ν, v) = f(v). Thus, there exists a solution ν. Now the stability estimate (1.37) is left to prove. Using the coercivity of our bilinear form yields a(u, u) f(u) |f(u)| αkukV ≤ = ≤ sup = kfkV ∗ . kukV kukV u∈V kukV

∞ 1 1.2.17 Lemma: Let aij, c ∈ L (Ω), bi ∈ C (Ω) such that there holds for a positive constant α

2 T d α|ξ| ≤ ξ A(x)ξ, ∀ξ ∈ R , (1.38) 0 ≤ c + ∇·b.

1 Then, the associated bilinear form is coercive and bounded on H0 (Ω), and thus the weak formulation has a unique solution.

1.2.18 Deﬁnition: A diﬀerential equation of second order in divergence form (1.34) such that (1.38) holds is called elliptic. The lower bound α is the ellipticity constant.

19 1.3 Fast Facts on Sobolev Spaces

1.3.1. While this section reviews some of the basic mathematical properties of Sobolev spaces, it suﬀers a bit from overly abstract mathematical arguments. While the strongly mathematically inclined reader might appreciate this, it is not really necessary for the remainder of this lecture, where we only need the basic results.

1.3.2 Notation: For a multi-index α = (α1, . . . , αd) with nonnegative integer αi and a function with sufficient differentiability, we define the derivative

α α1 αd ∂ f = ∂1 ··· ∂d f. The order of ∂α is X |α| = αi.

1.3.3 Deﬁnition: If for a given function u there exists a function w such that Z Z ∞ wϕ dx = − u∂iϕ dx, ∀ϕ ∈ C00 (Ω), (1.39) Ω Ω

then we deﬁne ∂iu := w as the distributional derivative (partial) of ∞ ∞ u with respect to xi. Here, C00 (Ω) is the space of all functions in C (Ω) with compact support in Ω. Similarly through integration by parts, we deﬁne distributional direc- tional derivatives, distributional gradients, ∂αu, etc. We call a distributional derivative weak derivative in Lp if it is a function in this space.

Remark 1.3.4. Formula (1.39) is the usual integration by parts. Therefore, whenever u ∈ C1 in a neighborhood of x, the distributional derivative and the usual derivative coincide.

Example 1.3.5. Let Ω = R and u(x) = |x|. Intuitively, it is clear that the distributional derivative, if it exists, must be the Heaviside function ( −1 x < 0 w(x) = (1.40) 1 x > 0.

The proof that this is actually the distributional derivative is left to the reader.

20 Example 1.3.6. For the derivative of the Heaviside function in (1.40), we ﬁrst observe that it must be zero whenever x 6= 0, since the function is continuously diﬀerentiable there. Now, we take a test function ϕ ∈ C∞ with support in the interval (−ε, ε) for some positive ε. Let w0(x) be the derivative of w. Then, by integration by parts

Z ε Z 0 Z ε w(x)ϕ0(x) dx = − w(x)0ϕ(x) dx − w(x)0ϕ(x) dx + 2ϕ(0) = 2ϕ(0), −ε −ε 0 since w0(x) = 0 under both integrals. Thus, w0(x) is an object which is zero everywhere except at zero, but its integral against a test function ϕ is nonzero. This contradicts our notion, that integrable functions can be changed on a set of measure zero without changing the integral. Indeed, w0 is not a function in the usual sense, and we write w0(x) = 2δ(x), where δ(x) is the Dirac δ- distribution, which is deﬁned by the two conditions

δ(x) = 0, ∀x 6= 0 Z 0 δ(x)ϕ(x) dx = ϕ(0), ∀ϕ ∈ C (R). R We stress that δ is not an integrable function, or a function at all.

1.3.7 Deﬁnition: The Sobolev space W k,p(Ω) is the space

k,p p α p W (Ω) = u ∈ L (Ω) ∂ u ∈ L (Ω)∀|α| ≤ k , (1.41)

where the derivatives are understood in weak sense. Its norm is deﬁned by

p p X α p kvkk,p = kvkk,p;Ω = k∂ vkLp(Ω). (1.42) |α|≤k

The following seminorm will be useful:

p p X α p |v|k,p = |v|k,p;Ω = k∂ vkLp(Ω). (1.43) |α|=k

1.3.8 Notation: We will use the notation

kvk0 = kvk0;Ω = kvkL2(Ω).

Accordingly, W 0,p(Ω) = Lp(Ω).

21 1.3.9 Corollary: There holds

W k,p(Ω) ⊂ W k−1,p(Ω) ⊂ · · · ⊂ W 0,p(Ω) = Lp(Ω) (1.44)

1.3.10 Deﬁnition: The Sobolev space Hk,p(Ω) is the completion of ∞ p C (Ω) with respect to the norm k·kk,p. In the case p = 2, we write Hk(Ω) = Hk,2(Ω).

1.3.11 Theorem (Meyers-Serrin):

Hk,p(Ω) =∼ W k,p(Ω)

Example 1.3.12. Functions, which are in W k,p(Ω) or not.

1 1. The function x/|x| is in H (B1(0)) if d = 3, but not if d = 2.

1.3.13 Deﬁnition: A bounded domain Ω ⊂ Rd is said to have Ck- k boundary or to be a C -domain, if there is a ﬁnite covering {Ui} of its k boundary ∂Ω, such that for each Ui there is a mapping Φi ∈ C (Ui) with the following properties:

d Φi(∂Ω ∩ Ui) ⊂ x ∈ R | x1 = 0 , (1.45) d Φi(Ω ∩ Ui) ⊂ x ∈ R | x1 > 0 . The domain is called Lipschitz, if such a construction exists with Lipschitz-continuous mappings.

1.3.14 Deﬁnition: We say that a normed vector space U ⊂ V is continuously embedded in another space V , in symbolic language

U,→ V, (1.46)

if the inclusion mapping U 3 x 7→ x ∈ V is continuous, that is, there is a constant c such that

kxkV ≤ ckxkU . (1.47)

If the spaces U and V consist of equivalence classes, the inclusion may involve choosing representatives on the left or on the right.

22 1.3.15 Theorem: Let Ω ⊂ Rd be a bounded Lipschitz domain. For the space W k,p(Ω) deﬁne the number

d s = k − p . (1.48)

Assume k1 ≤ k2 and p1, p2 ∈ [1, ∞). Then, if s1 ≥ s2, we have the continuous embedding

W k1,p1 (Ω) ,→ W k2,p2 (Ω). (1.49)

1.3.16 Lemma: Let Ω be a bounded Lipschitz domain in Rd. Then, there exists a constant c only depending on Ω, such that every function u ∈ H1(Ω) ∩ C1(Ω) admits the estimate

kukLp(∂Ω) ≤ ckukW 1,p(Ω). (1.50)

1.3.17 Theorem (Trace theorem): Let Ω be a bounded Lipschitz domain in Rd. Then, every function u ∈ W 1,p(Ω) has a well deﬁned trace γu ∈ Lp(∂Ω) and there holds

kγukLp(∂Ω) ≤ ckukW 1,p(Ω), (1.51)

with the same constant as in the previous lemma. We simply write

u|∂Ω = γu. (1.52)

Remark 1.3.18. The trace theorem guarantees that the imposition of Dirichlet boundary conditions on Sobolev functions is a reasonable operation. In partic- 1 1 ular, it ensures that H (Ω) and H0 (Ω) are indeed different spaces. The same 1 1 2 does not hold, if we complete C (Ω) and C0 (Ω) in L (Ω). Remarkably, we set out defining W 1,p(Ω) as a subset of Lp(Ω), which consists of functions “defined up to a set of measure zero”. Now it turns out, that functions in W 1,p(Ω) can have well-defined values on certain sets of measure zero. From the point of view of subsets of Lp(Ω), this is always to be understood by choosing representatives of the equivalence class. This is also the reason why we write “,→” instead of “⊂”.

23 1.3.19 Deﬁnition: A function f ∈ C0(Ω) is Hölder-continuous with exponent γ ∈ (0, 1], if there is a constant Cf such that

γ |f(x) − f(y)| ≤ Cf |x − y| ∀x, y ∈ Ω. (1.53)

In particular, for γ = 1, we obtain Lipschitz-continuity. We deﬁne the Hölder space Ck,γ of k-times continuouly diﬀerentiable functions such that all derivatives of order k are Hölder-continuous. The norm is |∂αu(x) − ∂αu(y)| kukCk,γ (Ω) = max sup γ (1.54) |α|≤k x,y∈Ω |x − y|

1.3.20 Theorem: Let Ω ⊂ Rd be a bounded Lipschitz domain. If d k,p s = k − p > j + γ, then every function in W has a representative in Cj,γ . We write

W k,p(Ω) ,→ Cj,γ (Ω). (1.55)

1.3.21 Corollary: Elements of Sobolev spaces are continuous if the derivative order is suﬃciently high. In particular,

H1(Ω) ,→ C(Ω) d = 1, (1.56) H2(Ω) ,→ C(Ω) d = 2, 3.

1.4 Regularity of Weak Solutions

1.4.1. So far, we have proven existence and uniqueness of weak solutions. We have seen, that these solutions may not even be continuous, far from diﬀeren- tiable. In this section, we collect a few results from the analysis of elliptic pde which establish higher regularity under stronger conditions.

k,p 1.4.2 Deﬁnition: The space Wloc (Ω) consists of functions u such that k,p u ∈ W (Ω1) for any Ω1 ⊂⊂ Ω, where the latter reads compactly em- k bedded, namely Ω1 ⊂ Ω. Similarly, we deﬁne Hloc.

24 1.4.3 Theorem ([Gilbarg and Trudinger, 1998, Theorem 8.8]): 0,1 ∞ 1 Let aij ∈ C (Ω) and bi, c ∈ L (Ω). If u ∈ H (Ω) is a solution to the 2 2 elliptic equation and f ∈ L (Ω), then u ∈ Hloc(Ω).

k,1 1.4.4 Theorem (Interior regularity): Let aij ∈ C (Ω) and bi, c ∈ Ck−1,1(Ω). If u ∈ H1(Ω) is a solution to the elliptic equation and f ∈ k,2 k+2 W (Ω), then u ∈ Hloc (Ω).

Proof. [Gilbarg and Trudinger, 1998, Theorem 8.10]

1.4.5 Corollary: If in the interior regularity theorem d = 2, 3, then u is a classical solution of the PDE if k ≥ 2. ∞ ∞ If aij, bi, c ∈ C (Ω), then u ∈ C (Ω).

1.4.6 Theorem (Global regularity): If in addition to the assumptions of the interior regularity theorem Ω is a Ck+2-domain, then the 1 solution u ∈ H0 (Ω) to the homogeneous Dirichlet boundary value problem problem is in Hk+2(Ω).

Proof. [Gilbarg and Trudinger, 1998, Theorem 8.13]

d 2 1.4.7 Corollary: Let Ω ⊂ R with d = 2, 3 be a C -domain, aij ∈ 0,1 ∞ 1 C (Ω) and bi, c ∈ L (Ω). If u ∈ H0 (Ω) is a solution to the elliptic equation and f ∈ L2(Ω), then u ∈ H2(Ω).

1.4.8 Remark: In order to guarantee a classical solution by these arguments, we must require that Ω has C4 boundary.

1.4.9 Remark: The condition ∂Ω ∈ C2 in the previous corollary can be replaced by the assumption that Ω is convex.

25 1.4.10 Theorem (Kondratev): Let the assumptions of the interior regularity theorem Theorem 1.4.3 hold. Assume further that ∂Ω is piecewise C2 with ﬁnitely many irregular points. Then, the solution 1 u ∈ H0 (Ω) of the elliptic PDE admits a representation

n X u = u0 + ui, (1.57) i=1

2 where u0 ∈ H (Ω) and ui is a singularity function associated with the irregular point xi.

Proof. [Kondrat’ev, 1967]

26 Chapter 2

Conforming Finite Element Methods

2.1 Meshes, shape functions, and degrees of freedom

2.1.1 Deﬁnition: Let T ⊂ Rd be a polyhedron. We call the lower dimensional polyhedra constituting its boundary facets. A facet of dimension zero is called vertex, of dimension one edge, and a facet of codimension one is called a face.

2.1.2 Deﬁnition: A mesh T is a nonoverlapping subdivision of the domain Ω into polyhedral cells denoted by T , for instance simplices, quadrilaterals, or hexahedra. The faces of a cell are denoted by F , the vertices by X. Cells are typically considered open sets. A mesh T is called regular, if each face F ⊂ ∂T of the cell T ∈ T is either a face of another cell T 0, that is, F = T ∩ T 0, or a subset of ∂Ω.

Remark 2.1.3. For this introduction, we will assume that indeed Ω is the union of mesh cells, which means, that its boundary consists of a ﬁnite union of planar faces. The more general case of a mesh approximating the domain will be deferred to later discussion.

27 2.1.4 Deﬁnition: With a mesh cell T , we associate a ﬁnite dimensional shape function space P(T ) of dimension nT . The term node functional denotes linear functionals on this space. i A set of node functionals {NT }i=1,...,nT is called unisolvent on P(T ) if T for any vector u = (u1, . . . , unT ) there exists a unique u ∈ P(T ) such that

i NT (u) = ui, i = 1, . . . , nT . (2.1)

A ﬁnite element is a set of shape function spaces P(T ) for all T ∈ T together with unisolvent set of node functionals.

2.1.5 Notation: If the node functionals N i are unisolvent on P(T ), then, there is a basis {pk} of P(T ) such that

i N (pk) = δik. (2.2)

We refer to {pk} as shape function basis and use the term degrees of freedom for both the node functionals and the basis functions.

2.1.6 Deﬁnition: Node functionals can be associated with the cell T or with one of its lower dimensional boundary facets. We call this asso- ciation the topology of the ﬁnite element.

2.1.7 Deﬁnition: The ﬁnite element space on the mesh T, denoted by VT is a subset of the concatenation of all shape function spaces, 2 VT ⊂ f ∈ L (Ω) f|T ∈ P(T ) . (2.3)

The degrees of freedom of VT are the union of all node functionals, where we identify node functionals associated to boundary facets among all cells sharing this facet. The resulting dimension is X n = dim VT ≤ nT . (2.4)

2.1.8 Notation: When we enumerate the degrees of freedom of VT, we obtain a global numbering of degrees of freedom N i with i = 1, . . . , n. j For each mesh cell, we have a local numbering NT with j = 1, . . . , n. By construction of the ﬁnite element space, there is a unique i, such that j i NT (f) = N (f) for all cells T and local indices j. The converse is not true due to the identiﬁcation process.

28 • • • • • • • • • • • •

• •

Figure 2.1: Identification of node functionals. The node functionals on shared edges (separated for presentation purposes) are distinguished locally as belong- ing to their respective cells, but identical global indices are assigned to all nodes in a single circle. Thus, all associated shape functions obtain the same coefficient in the global basis representation of a finite element function u.

i j 2.1.9 Deﬁnition: We refer to the mapping between N and NT as the mapping between global and local indices

ι :(T, j) 7→ i. (2.5)

It induces a “natural” basis {vi} of VT by

vi|T = pT,j, (2.6)

i where {pT,j} is the shape function basis on T . For each N , we deﬁne T(N i) as the set of cells T sharing the node functional N i, and

i [ Ω N = T. (2.7) i T ∈T(N )

2.1.10 Lemma: The support of the basis function vi ∈ VT is i supp(vi) ⊂ Ω N .

2.1.11 Lemma: Let T be a subdivision of Ω, and let u be a function 1 on Ω, such that u|T ∈ C (T ) for each T ∈ T. Then,

u ∈ H1(Ω) ⇐⇒ u ∈ C(Ω). (2.8)

29 2.1.12 Lemma: We have VT ⊂ C(Ω) if and only if for every facet F of dimension dF < d there holds that

1. the traces of the spaces P(T ) on F coincide for all cells T having F as a facet, 2. The node functionals associated to the facet are unisolvent on this trace space.

2.1.1 Shape function spaces on simplices

d 2.1.13 Deﬁnition: A simplex T ∈ R with vertices X0,..., Xd is de- T scribed by a set of d + 1 barycentric coordinates λ = (λ0, . . . , λd) such that

0 ≤ λi ≤ 1 i = 0, . . . , d; (2.9)

λi(Xj) = δij i, j = 0, . . . , d (2.10) X λi(x) = 1, (2.11)

and there holds n o d X T = x ∈ R x = Xkλk . (2.12)

d+1×d 2.1.14 Lemma: There is a matrix BT ∈ R and a vector bT ∈ Rd+1, such that

λ = BT x + bT . (2.13)

2.1.15 Corollary: The barycentric coordinates λ0, . . . , λd are the linear Lagrange interpolating functions for the points X0,..., Xd. In particular, λk ≡ 0 on the facet not containing Xk.

Example 2.1.16. We can use barycentric coordinates to deﬁne interpolating polynomials on simplicial meshes easily, as in Table 2.1.

Remark 2.1.17. The functions λi(x) are the shape functions of the linear P1 element on T . They allow us to deﬁne basis functions on the cell T without use of a reference element Tb.

Note that λi ≡ 0 on the face opposite to the vertex xi.

30 Degrees of freedom Shape functions

ϕi = λi, i = 0, 1, 2

2 ϕii = 2λi − λi, i = 0, 1, 2

ϕij = 4λiλj j 6= i

1 ϕiii = 2 λi(3λi − 1)(3λi − 2) i = 0, 1, 2 9 ϕij = 2 λiλj(3λj − 1) j 6= i ϕ0 = 27λ0λ1λ2

Table 2.1: Degrees of freedom and shape functions of simplicial elements in terms of barycentric coordinates

2.1.2 Shape functions on tensor product cells

2.1.18 Deﬁnition: The space of tensor product polynomials of degree k in d dimensions, denoted as Qk consists of polynomials of degree up to k in each variable. Given a basis for one-dimensional polynomials {pi}i=0,...,k, a natural basis for Qk is the tensor product basis

d Y pi1,...,id (x) = pi1 ⊗ · · · ⊗ pid (x) = pik (xk). (2.14) k=1

Remark 2.1.19. Note that the basis functions of Qk can be denoted as products of univariate polynomials, but that general polynomials in this space as linear combinations of these basis functions do not have this structure.

31 2.1.20 Lemma: Let {Nj} be a set of one-dimensional node functionals dual to the one-dimensional basis {pi} such that

Nj(pi) = δij. (2.15)

Then, a dual basis for {pi1,...,id } is obtained by deﬁning on the tensor product basis of Qk

d Y Nj1,...,jd (pi1,...,id ) = Nj1 ⊗ · · · ⊗ Njd (pi1 ⊗ · · · ⊗ pid ) = Njk (pik ). k=1 (2.16)

Proof. It is a theorem in linear algebra, that a linear functional on a vector space is uniquely deﬁned by its values on a basis of the space. Thus, (2.16) uniquely deﬁnes the node functionals Nj1,...,jd . The duality property follows from the fact that

d Y Nj1,...,jd (pi1,...,id ) = δik,jk , k=1 which is one if and only if all index pairs match and zero in all other cases.

Example 2.1.21. Let a basis {pi} of the univariate space Pk be deﬁned by Lagrange interpolation in k + 1 points tj ∈ [0, 1]. A basis of the d-dimensional space Qk is then obtained by all possible products

d Y pi1,...,id (x) = pik (xk). k=1 The node functionals following the construction above are obtained by

d Y Nj1,...,jd (pi1,...,id ) = pik (xjk ). k=1 Finally, we have to convert the term on the right into an expression, which can be applied to any polynomial in Qk. To this end, we observe that

d Y pik (xjk ) = pi1,...,id (xj1 , . . . , xjd ). k=1 Therefore, we conclude that the tensor product node functionals resulting from this construction are

Nj1,...,jd (p) = p(xj1 , . . . , xjd ).

32 2.1.22 Example (The space Q2):

2.1.23 Lemma: The trace of the d-dimensional tensor product polynomial space Qk on the δ-dimensional facets of the reference cube d Tb = (0, 1) is the δ-dimensional space Qk. The traces from two cells sharing the same face coincide, if the mapping is continuous. Therefore, continuity can be achieved by unisolvent sets of node functionals on the face.

Proof. By keeping d − δ variables constant in the tensor product basis in (2.14).

33 2.1.24 Example (Continuous basis functions):

2.1.25 Example (Discontinuous basis functions):

Example 2.1.26. As a second example, we choose d = 2 and the univariate space P2 with node functionals Z 1 N0(p) = p(0), N1(p) = p(t) dt, N2(p) = p(1), (2.17) 0 that is, a mixture of Lagrange interpolation and orthogonality on the interval

34 [0, 1]. The matching basis polynomials are 2 2 p0(t) = 3(1 − t) − 2(1 − t), p1(t) = 6t(1 − t), p2(t) = 3t − 2t. (2.18) Follwoing the construction of the previous example, we obtain

N00(p) = p(0, 0), N02(p) = p(0, 1), (2.19) N20(p) = p(1, 0), N22(p) = p(1, 1). Then, Z 1 Z 1 N01(p01) = N01(p0 ⊗ p1) = p0(0) p1(y) dy = p(0, y) dy. (2.20) 0 0

Thus, the node functional N01 is the integral over the left edge of the reference square. By the same construction, N01 is the integral over the right edge. N10 and N12 are the integrals over the bottom and top edge, respectively. Finally, Z 1 Z 1 Z 1 Z 1 N11(p11) = p1(x) dx p1(y) dy = p11(x, y) dx dy. (2.21) 0 0 0 0 Thus, the tensor product of two line integrals becomes the integral over the area.

2.1.3 The Galerkin equations and Céa’s lemma

2.1.27 Deﬁnition (Galerkin approximation): Let u ∈ V be determined by the weak formulation

a(u, v) = f(v) ∀v ∈ V,

where V is a suitable function space including boundary conditions. The Galerkin approximation, also called conforming approximation of this problem reads as follows: choose a subspace Vn ⊂ V of dimension n and ﬁnd un ∈ Vn, such that

a(un, vn) = f(vn) ∀vn ∈ Vn.

We will refer to this equation as the discrete problem.

2.1.28 Corollary (Galerkin equations): After choosing a basis {vi} for Vn, the Galerkin equations are equivalent to a linear system

Au = f, (2.22)

with A ∈ Rn×n and f ∈ Rn deﬁned by

aij = a(vj, vi), fi = f(vi). (2.23)

35 2.1.29 Lemma: If the lemma of Lax-Milgram holds for a(., .) on V , it holds on Vn ⊂ V . In particular, solvability of the Galerkin equations is implied.

2.1.30 Lemma (Céa): Let a(., .) be a bounded and elliptic bilinear form on the Hilbert space V . Let u ∈ V and un ∈ Vn ⊂ V be the solution to the weak formulation and its Galerkin approximation

a(u, v) = f(v) ∀v ∈ V,

a(un, vn) = f(vn) ∀vn ∈ Vn, respectively. Then, there holds M ku − uhkV ≤ inf ku − vhkV . (2.24) α vn∈Vn

2.1.31 Lemma: For a ﬁnite element discretization of Poisson’s equation with the space VT, the Galerkin equations can be computed using the following formulas:

Z Z X Z aij = ∇vj · ∇vi dx = ∇vj · ∇vi dx = ∇vj · ∇vi dx T ∈ ( i) Ω Ω(N i) T N T Z Z X Z fi = fvi dx = fvi dx = fvi dx T ∈ ( i) Ω Ω(N i) T N T

2.1.32 Algorithm (Assembling the matrix):

1. Start with a matrix A = 0 ∈ Rn×n 2. Loop over all cells T ∈ T

nT ×nT 3. On each cell T , compute a cell matrix AT ∈ R by integrating Z aT,ij = ∇pT,j · ∇pT,i dx, (2.25) T

where {pT,i} is the shape function basis. 4. Assemble the cell matrices into the global matrix by

aι(i),ι(j) = aι(i),ι(j) + aT,ij i, j = 1, . . . , nT . (2.26)

36 2.1.4 Mapped ﬁnite elements

2.1.33 Deﬁnition: A mapped mesh T is a set of cells T , which are deﬁned by a single reference cell Tb and individual smooth mappings

d ΦT : Tb → R (2.27) ΦT (Tb) = T. The deﬁnition extends to small sets of reference cells, for instance for triangles and quadrilaterals.

2.1.34 Example: Let the reference triangle Tb be deﬁned by xb Tb = x,b yb > 0, xb + yb < 1 . (2.28) yb

Then, every cell T spanned by the vertices X0, X1, and X2 is obtained by mapping Tb by the aﬃne mapping X1 − X0 X2 − X0 xb X0 ΦT (xb) = + =: BT xb + bT (2.29) Y1 − Y0 Y2 − Y0 yb Y0

2.1.35 Example: The reference cell for a quadrilateral is the reference 2 square Tb = (0, 1) . Every quadrilateral T spanned by the vertices X0 to X3 is then obtained by the bilinear mapping

ΦT (xb) = X0(1 − xb)(1 − yb) + X1xb(1 − yb) + X2(1 − xb)yb + X3xbyb (2.30)

2.1.36 Deﬁnition: Mapped shape functions {pi} on a mesh cell T are deﬁned by a set of shape functions {pbi} on the reference cell Tb through pull-back

−1 pi(x) = pi Φ (x) = pi(x), b b b (2.31) −T ∇pi(x) = ∇Φ (xb)∇b pbi(xb)

37 2.1.37 Lemma: Let Tb be the reference triangle and let T be a triangular mesh cell with mapping x = ΦT (xb) = Bxb + b. Let there hold u(x) = k k ub(xb). Then, u ∈ H (T ) if and only if ub ∈ H (Tb) and we have with some constant c the estimates

|u| ≤ ckBkk(det B)−1/2|u| , b k,Tb k,T (2.32) |u| ≤ ckB−1kk(det B)1/2|u| . k,T b k,Tb

2.1.38 Lemma: For a cell T , let R be the radius of the circumscribed circle and % the radius of the inscribed circle. Then,

kBk ≤ cR, kB−1k ≤ c%−1. (2.33)

2.1.39 Assumption: For more general mappings Φ: Tb → T , we make the assumption, that they can be decomposed into three factors,

Φ = ΦO ◦ ΦS ◦ ΦW , (2.34)

where ΦO is a combination of translation and rotation, ΦS is a scaling with a characteristic length hT , and ΦW is a warping function not changing the characteristic length.

Example 2.1.40. We construct the inverse of Φ in two dimensions by the following three steps, using as hT the length of the longest edge of T .

1. Choose ΦO as the rigid body movement which maps the longest edge to the interval (0, hT ) on the x-axis and the cell itself to TO in the positive half plane. This mapping has the structure

−1 ΦO (x) = Sx − SX0,

where S is an orthogonal matrix and X0 is the vertex moved to the origin. 2. Choose the scaling

Φ−1(x) = 1 x, S hT

such that the longest edge of the resulting cell TS has the longest edge equal to the interval (0, 1) on the x-axis.

−1 3. Warp the cell TS into the reference cell Tb by the mapping ΦW . This operation leaves the longest edge untouched. For triangles, it is the uniquely deﬁned linear transformation mapping the vertex not on the longest edge to (0, 1). For quadrilaterals, it is a bilinear transformation.

38 In the ﬁrst step, we have assumed that the cell is convex, which is always true for triangles. For nonconvex quadrilaterals, it can be shown that the determi- nant of ∇Φ changes sign inside the cell, such that these cells are not useful for computations.

The idea of this decomposition is, that we separate mappings changing the position, size, and shape of the cells.

2.1.41 Lemma (Scaling lemma): Let the typical length of a cell T be hT . Assume there are constants 0 < MT , mT , dT ,DT , such that

k∇ΦW (xb)k ≤ MT , −1 −1 k∇ΦW (xb)k ≤ mT , (2.35) 2 2 dT ≤ det ∇ΦW (xb)) ≤ DT .

for all xb ∈ Tb. Then, for k = 0, 1 and a constant c

M d |u| ≤ c T hk− /2|u| , b k,Tb d T k,T T (2.36) D d |u| ≤ c T h /2−k|u| . k,T T b k,Tb mT This extends to higher derivatives under assumptions on higher derivatives of ΦT .

Proof. By the chain rule, ∇ΦT = ∇ΦO∇ΦS∇ΦW . By construction, ∇ΦO is an orthogonal matrix, such that

−1 k∇ΦOk = k∇ΦO k = 1.

Since it preserves angles and lengths, det ∇ΦO = 1. Since ΦS is a multiple of the identity, we have

−1 1 d k∇ΦSk = hT , k∇ΦS k = , det ∇ΦS = hT . hT By change of variables, we have Z Z Z 2 2 2 u dx = ub |det ∇ΦT | dxb = ub det ∇ΦS det ∇ΦO det ∇ΦW dxb, T Tb Tb such that the case k = 0 is proven immediately by Z Z Z d 2 2 2 d 2 2 hT dT ub dxb ≤ u dx ≤ hT DT ub dxb Tb T Tb

39 By the chain rule, we have

T T T T ∇b ub(xb) = ∇Φ ∇u(x) = ∇ΦW ∇ΦS ∇ΦO∇u(x), such that there holds

|∇b u(x)| ≤ k∇ΦW khT |∇u|, b b (2.37) −1 −1 |∇u(x)| ≤ k∇ΦW khT |∇b ub|.

2.1.42 Remark: We have dT = DT , if and only if the mapping is aﬃne. The quotient MT /mT measures how much the shape of the mesh cell deviates from the reference cell. For instance, it is one for squares.

2.2 A priori error analysis

2.2.1. In this section, we develop error estimates of the following type

If the size of mesh cells converges to zero, then the diﬀerence between the true solution and the ﬁnite element solution converges to zero with a certain order.

They are thus a prediction, that the solutions actually converge, and they measure the asymptotic convergence rate. They do contain unknown constants, such that they are no prediction of the error on a given mesh.

The theory in this chapter is about bilinear forms which are bounded and elliptic on a subspace V ⊂ H1(Ω) determined by boundary conditions. We will choose 1 V = H0 (Ω), even if more general boundary conditions are possible. It is a good approach to think of the Dirichlet problem for the Laplacian, even if we allow for somewhat more general equations.

40 2.2.1 Approximation of Sobolev spaces by ﬁnite elements

2.2.2 Lemma (Poincaré inequality): Let Ω be a bounded Lipschitz domain. For any function u ∈ H1(Ω) deﬁne

1 Z u = u(x) dx, (2.38) |Ω| Ω where |Ω| denotes the measure of Ω. There exists a constant c depending on the domain only, such that each of the following inequalities hold:

ku − ukL2(Ω) ≤ ck∇ukL2(Ω) (2.39) 2 2 2 kukL2(Ω) ≤ c k∇ukL2(Ω) + u (2.40)

Proof. The proof exceeds the tools we have developed in this class. The proof in [Gilbarg and Trudinger, 1998, Section 7.8] seems elementary and direct, but is technical and requires star-shaped domains. The proof in [Evans, 1998, Section 5.8.1] is more elegant, but it uses compact embedding and is indirect, such that the constant cannot be determined.

2.2.3 Lemma (Bramble-Hilbert): Let T ⊂ Rd be a domain with Lipschitz boundary and let s(.) be a bounded sublinear functional on Hk+1(T ) with the property

s(p) = 0 ∀p ∈ Pk. (2.41) Then, there exists a constant c only dependent on T such that

|s(v)| ≤ c|v|k+1,T . (2.42)

k+1 Proof. Since s(·) is sublinear and vanishes on Pk, we have for v ∈ H (T ):

|s(v)| ≤ |s(v + p)| + |s(p)| = |s(v + p)| ∀p ∈ Pk. (2.43) We will construct a polynomial, such that 1 Z ∂α(v + p) = ∂α(v + p) dx = 0 ∀|α| ≤ k, (2.44) |T | T that is, the sum v + p and all its derivatives up to order k are mean-value free. Thus, by recursive application of Poincaré inequality, we get for |α| ≤ k

2 h 2 2i 2 kv + pkL2(T ) ≤ c k∇(v + p)kL2(T ) + v + p ≤ c|v + p|1;T h 2i α 2 α 2 α 2 k∂ (v + p)kL2(T ), ≤ c k∇∂ (v + p)kL2(T ) + ∂ (v + p) ≤ c|v + p||α|+1;T ,

41 such that kv + pkk+1;T ≤ |v + p|k+1;T . Furthermore, since p ∈ Pk

α α k∂ (v + p)kL2(T ) = k∂ vkL2(T ) ∀|α| = k + 1.

Combining with (2.43), we obtain

|s(v)| ≤ c|v + p|k+1;T ≤ c|v|k+1;T .

It remains to construct the polynomial p with the desired properties. To this end, we note that for two multi-indices α and β holds that ∂αxβ = 0 as soon as αi > βi for some index i. Let

X β p(x) = aβx . |β|≤k

Then, for any |α| = k we get

α ∂ p(x) = α! δαβ, where α! = α1!α2! . . . αd!. Thus, we can use condition (2.44) to ﬁx the coeﬃ- cients aβ to Z 1 β aβ = ∂ v dx |β| = k. β! |T | T

Thus, we have decomposed p =p ˜k + pk−1, where p˜k is known and pk−1 ∈ Pk−1. Thus, we can repeat the process determining the coeﬃcients of highest order in pk−1, Z 1 β aβ = ∂ (v − p˜k) dx |β| = k − 1. β! |T | T Recursion down to k = 0 yields the polynomial p with the desired property.

k+1 2.2.4 Corollary: Let Π: H (T ) → Pk be a continuous, linear projector. For any m ≤ k there exists a constant c such that

ku − Πukm,T ≤ c|u|k+1,T . (2.45)

42 2.2.5 Deﬁnition: Let {Th} for h > 0 be a family of meshes parametrized by the parameter

h = max hT , (2.46) T ∈Th where hT is the characteristic length from the discussion of mappings, for instance the diameter of T . Such a family is called shape regular, if the constants MT , mT , dT , and DT in the scaling lemma can be chosen independent of the cell T ∈ Th and of h > 0. The family is called quasi- uniform, if in addition there is a positive constant independent of h such that

h ≤ c min hT . (2.47) T ∈Th

2.2.6 Definition: Let V be a function space on Ω, and let Vh = VTh be a finite element space on the mesh Th on Ω with node functionals Ni and dual basis pi, where i = 1, . . . , nh. We define the nodal interpolation operator by

Ih : V → Vh X (2.48) v 7→ Ni(v)pi.

2.2.7 Lemma: The nodal interpolation operator Ih is a projector. It is continuous on H2(Ω) if the dimension is d = 2, 3 and the node functionals are deﬁned as Lagrange interpolation.

2.2.8 Deﬁnition: On a mesh Th, we deﬁne the broken Sobolev norm and seminorm by

2 X 2 kukk;h = kukk;T T ∈ Th (2.49) 2 X 2 |u|k;h = |u|k;T T ∈Th

43 2.2.9 Theorem: Let {Th} be a shape regular family of meshes. Let the ﬁnite element spaces Vh = VTh . Let the nodal interpolation operator Ih k+1 be surjective onto Pk on every cell T ∈ Th and continuous on H (Ω). Then, there is a constant c such that for any u ∈ Hk+1(Ω) and m ≤ k+1 there holds

k+1−m ku − Ihukm;h ≤ ch |u|k+1;h. (2.50)

Proof. We have by deﬁnition X |u − Ihu|m;h = |u − Ihu|m;T . T ∈Th Using the scaling lemma, we get

d/2−m |u − Ihu|m;T ≤ chT |ub − Idhu|. On the reference cell, we use the Bramble-Hilbert lemma, more precisely, Corol- lary 2.2.4 to obtain

|u − I u| ≤ c|u| . b dh b k+1;Tb Scaling back yields

d |u| ≤ chk+1− /2|u| . b k+1;Tb T k+1;T Combining, we obtain

k+1−d/2−m+d/2 |u − Ihu|m;T ≤ chT |u|k+1;T .

k+1−m Summing up and pulling the maximum of hT out of the sum yields the result for hT ≤ 1.

Remark 2.2.10. Strictly speaking, we have only proven the result for hT ≤ 1. But then, if diam Ω = 1, this condition is always true. Therefore, by rescaling the domain before computing, the estimate holds in general.

2.2.11 Corollary: Let a(., .) be a bounded and elliptic bilinear form 1 on V = H0 (Ω) and let the finite element space Vh be defined on a shape-regular family of meshes {Th}, such that the interpolation estimate eq. (2.50) holds. If furthermore the solution u ∈ Hk+1(Ω), the error of the finite element solution uh ∈ Vh ⊂ V admits the estimate

k ku − uhk1;h ≤ ch |u|k+1;h. (2.51)

44 2.2.12. For 2nd order elliptic problems, we have now derived estimates of the H1-norm of the error under the assumption that the solution exhibits further regularity. Now, let us drop this assumptionfor an assumption on the boundary condition only, to obtain a qualitative convergence result.

2.2.13 Theorem: Let a(., .) be a bounded and elliptic bilinear form 1 on V = H0 (Ω) and let the finite element space Vh be defined on a shape-regular family of meshes {Th}, such that the interpolation estimate eq. (2.50) holds. Let u ∈ V and uh ∈ Vh be solutions to the exact and the finite element versions of a 2nd order boundary value problem. Then,

lim ku − uhk1;Ω = 0. (2.52) h&0

2.2.2 Estimates of stronger norms

2.2.14. So far, we have seen error estimates in a “natural norm” deﬁned as a norm such that the Lax-Milgram lemma holds for a given bilinear form a(., .). In this subsection, we now consider the question of estimates in stronger norms, such that the bilinear form is not elliptic with respect to this norm.

2.2.15 Deﬁnition: Let k·kX and k·kY be norms on a vector space V . We call k·kX a stronger norm than k·kY , if there is a constant c such that for all v ∈ V :

kvkY ≤ ckvkX . (2.53)

In this case, k·kY is called the weaker norm. If a converse inequality holds, the norms are called equivalent.

Example 2.2.16. For a bounded domain Ω, the norms k·kk+1 and k·kk are 1 both deﬁned on V = H0 (Ω). By the Sobolev embedding theorem, there is a constant c such that for any v ∈ V

kvkk ≤ ckvkk+1

2.2.17 Lemma (Inverse estimate): Let T be a mesh cell of size hT . Then, there is a constant only depending on k and the constants of the scaling lemma, such that for every u ∈ Pk there holds

−1 kuk1;T ≤ chT kuk0;T . (2.54)

45 2.2.18 Theorem: Let a(., .) be a bounded and elliptic bilinear form 1 on V ⊂ H (Ω) and let {Th} be a family of quasi-uniform meshes with ﬁnite element spaces Vh ⊂ V containing the space Pk with k ≥ 2 on each mesh cell. If furthermore the solution u ∈ Hk+1(Ω), the error between the exact and the ﬁnite element solution to a uniquely solvable elliptic boundary value problem, admits the estimate

k−1 ku − uhk2;h ≤ ch |u|k+1;h. (2.55)

Proof. We cannot apply Céa’s lemma directly, since the bilinear form is not elliptic with respect to the broken H2-inner product on the space H1(Ω). On the other hand, the error is not polynomial, such that we cannot apply the inverse estimate to it. Instead, we use triangle inequality

ku − uhk2;h ≤ ku − Ihuk2;h + kIhu − uhk2;h, and observe, that we already have the desired estimate for the ﬁrst term. For the second, we estimate by inverse estimate on each cell

2 min hT kIhu − uhk2;hkIhu − uhk1 ≤ ckIhu − uhk1, T ∈Th and c kI u − u k2 ≤ a(I u − u ,I u − u ) h h 1 γ h h h h c = a(I u − u, I u − u ) γ h h h cM ≤ kI u − uk kI u − u k γ h 1 h h 1 k ≤ Ch |u|k+1;hkIhu − uhk1, where we have used Galerkin orthogonality and the interpolation estimate. Combining the two and using the quasi-uniformity, we obtain the result of the theorem.

2.2.3 Estimates of weaker norms and linear functionals

2.2.19. Deriving optimal error estimates in the L2-norm cannot be achieved by the same technique as used for the H2-norm, as the following simple argument shows: using triangle inequality, we obtain

ku − uhkL2 ≤ ku − IhukL2 + kIhu − uhkL2 , we obtain that the ﬁrst term is of order hk+1. Thus, for an optimal error estimate, we require that the second term be of order hk+1 as well. We need

46 a replacement of the inverse estimate, which gains an order of h instead of loosing it. This is Poincaré inequality, but it requires an almost mean-value free function. And since Ihu is not the interpolant of uh, we cannot guarantee a small mean value of the diﬀerence on each mesh cell.

To the rescue comes a “duality argument” known as Aubin-Nitsche trick, which we introduce now.

2.2.20 Definition: Let the weak form of a boundary value problem on the domain Ω be defined as: find u ∈ V such that

a(u, v) = f(v) ∀v ∈ V.

Then, the dual problem, also called adjoint problem, with right hand side g ∈ V ∗ is: ﬁnd u∗ ∈ V such that

a(v, u∗) = g(v) ∀v ∈ V.

2.2.21 Lemma: The adjoint problem of the Dirichlet boundary value problem for Poisson’s equation is equal to the dual problem, that is, for 1 u ∈ H0 (Ω), the two statements

a(u, v) = f(v) ∀v ∈ V, a(v, u) = f(v) ∀v ∈ V,

are equivalent.

2.2.22 Assumption (Elliptic regularity): Let a(., .) be a bounded 1 and elliptic bilinear form on H0 (Ω). We say that a boundary value problem has elliptic regularity, if for any g ∈ L2(Ω) the solution u is in H2(Ω). In other words, there is a constant c independent of g, such that

kuk2;Ω ≤ ckgk0;Ω. (2.56)

Example 2.2.23. By Remark 1.4.9, a second order PDE with coeﬃcients aij ∈ 0,1 ∞ C (Ω) and bi, c ∈ L (Ω) has elliptic regularity. The same holds by integration by parts for ints adjoint.

The same boundary value problem does not have elliptic regularity, if the domain has a nonconvex corner, since we have corner singularity functions which are not in H2(Ω), and we can construct a right hand side g ∈ L2(Ω), which produces such singularities.

47 2.2.24 Theorem: Let a(., .) be a bounded and elliptic bilinear form on 1 V = H0 (Ω) and let the finite element space Vh be defined on a shape- regular family of meshes {Th}, such that the interpolation estimate (2.50) holds. If furthermore the solution u ∈ Hk+1(Ω) and the dual problem has elliptic regularity, the error of the finite element solution uh ∈ Vh ⊂ V admits the estimate

k+1 ku − uhk0 ≤ ch |u|k+1;h. (2.57)

2.2.25 Corollary: Under the assumptions of Theorem 2.2.24, let J(.) be a bounded linear functional on L2(Ω). Then,

k+1 |J(u) − J(uh)|0 ≤ ch |u|k+1;h. (2.58)

2.2.4 Green’s function and maximum norm estimates

2.2.26. In this section, we will introduce the basic concepts needed for pointwise error estimation. They heavily rely on Green’s function, which is useful for a general understanding of the solution structure as well.

2.2.27 Definition: For a differential equation Lu = f in Ω with boundary conditions u = 0 on the boundary ∂Ω, we define Green’s function G(y, x) associated to the point y ∈ Ω as solution to the problem

LG(y, x) = δ(x − y) ∀x ∈ Ω, (2.59) G(y, x) = 0 ∀x ∈ ∂Ω.

2.2.28 Theorem: Green’s function for Poisson’s equation on the whole space Ω = Rd is

( 1 − 2π log|x − y| d = 2 G(y, x) = 1 1 (2.60) d−2 d ≥ 3. d(d−2)|B1(0)| |x−y|

Proof. See [Evans, 1998, Section 2.2].

48 2.2.29 Lemma: Let Ω ⊂ Rd be a bounded domain. Then, Green’s function associated to a point y ∈ Ω for a linear diﬀerential operator L on Ω is obtained as the sum

G(y, x) = G∞(y, x) − G0(y, x), (2.61)

where G∞(y, x) is Green’s function for the whole space and G0(y, x) solves LG0(y, x) = 0 with boundary values G∞(y, x). If the domain is 2 convex, then, G0(y,.) ∈ H (Ω) for any interior point y.

2.2.30 Theorem: Let f ∈ C(Ω) and let Ω be such that the solution to

Lu = f in Ω, (2.62) u = 0 on ∂Ω,

is in C2(Ω). Then, Z u(x) = f(y)G(y, x) dy. (2.63) Ω

Proof. See [Evans, 1998, Section 2.2]

2.2.31 Theorem: Let a(., .) be a bounded and elliptic bilinear form 1 2 on V = H0 (Ω) on a bounded, convex domain Ω ⊂ R . Let the finite element space Vh be defined on a quasi-uniform family of meshes {Th} 2,∞ with local spaces P1 or Q1. If furthermore the solution u ∈ W (Ω), the error of the finite element solution uh ∈ Vh ⊂ V admits the estimate

2 ku − uhk∞ ≤ ch (1 + |log h|)|u|2,∞. (2.64)

Proof. The proof relies on solving a dual problem for the error in a single point. The solution is Green’s function for the adjoint equation, which is very irregular at the point of interest. Then, complicated analysis is needed to generate useful approximation estimates in spite of the singularity. Details can be found in [Schatz and Wahlbin, 1977, Rannacher and Scott, 1982].

Remark 2.2.32. This estimate extends to Ω ∈ Rd for d ≥ 3 and to higher polynomial degrees without the logarithmic factor. The regularity assumptions are quite high then.

49 2.3 A posteriori error analysis

2.3.1 Deﬁnition: Let u ∈ V be the solution to a boundary value in weak form and uh ∈ Vh be its ﬁnite element approximation on the mesh Th. We call a quantity ηh(uh) a posteriori error estimator,

ku − uhk ≤ cηh(uh). (2.65)

The estimator is reliable, if the constant c is computable. It is eﬃcient, if the converse estimate holds, namely

ηh(uh) ≤ cku − uhk. (2.66)

2.3.1 Quasi-interpolation in H1

2.3.2. Interpolation in Sobolev spaces in Theorem 2.2.9 relies on the nodal interpolation operator in Deﬁnition 2.2.6, which in turn requires point values of the interpolated function. Therefore, it is not deﬁned on H1(Ω) in dimensions greater than one. Since we need such interpolation operators in the analysis of a posteriori error estimates, we provide them in this section.

Most details in this section are from [Verfürth, 2013]. For the ease of presentation, we present the results for Dirichlet problem of the Laplacian, namely 1 V = H0 (Ω) and Z a(u, v) = ∇u · ∇v dx. Ω

2.3.3 Deﬁnition: A shape-regular family of meshes {Th} is called locally quasi-uniform, if there is a constant c such that for every pair of cells T1 and T2 sharing at least one vertex there holds

hT1 ≤ chT2 . (2.67)

50 2.3.4 Deﬁnition: For a vertex or higher-dimensional boundary facet F , we deﬁne the set of cells TF = T ∈ T F ⊂ ∂T . (2.68)

Similarly, the set of cells sharing at least one vertex with T is called TT . Additionally, we deﬁne the subdomains

[ [ 0 ΩF = T, ΩT = T . (2.69) 0 T ∈TF T ∈TT

2.3.5 Theorem (Clément quasi-interpolation): Let {Th} be a locally quasi-uniform family of meshes with piecewise polynomial ﬁnite 1 element spaces Vh ⊂ H0 (Ω). Then, there exist bounded operators 1 1 Ih : H0 (Ω) → Vh such that for every function u ∈ H0 (Ω), every mesh cell T , every face F , and for m = 0, 1 there holds

1−m ku − Ihukm;T ≤ chT |u|1;ΩT (2.70) 1/2 ku − Ihuk0;F ≤ chT |u|1;ΩT (2.71)

Proof. We construct the quasi-interpolation operator on a mesh cell T into the lowest order space P1 or Q1 in two steps. First, for each vertex Xi of T , let 1 Z ui = u dx. (2.72) |ΩX | i ΩXi By Poincaré inequality,

ku − u k ≤ c diam(Ω )k∇uk . (2.73) i ΩXi Xi ΩXi

For a vertex Xi on the boundary, we observe

1 Z Z 2 ku k2 = u dy dx i ΩX 2 i |ΩX | i ΩXi 1 Z 2 = 1u dy |Ω | Xi (2.74) 1 Z Z ≤ 12 dy u2 dy |ΩXi | = kuk2 ΩXi 2 2 ≤ c diam(ΩXi ) k∇uk , ΩXi

by Friedrichs’ inequality, since the subdomain ΩXi has at least one face on ∂Ω.

51 1 Now, we deﬁne the quasi-interpolation operator Ih : H0 (Ω) → Vh cellwise on simplices by X Ihu|T = λiui. (2.75)

Xi∈∂T ∩Ω Note, that zero boundary conditions are enforced by omitting vertices on the boundary. On quadrilaterals, we use the bilinear shape functions associated with P the vertices instead of the barycentric coordinates λi. Now, using λi = 1 and 0 ≤ λi ≤ 1, we estimate X X ku − IhukT ≤ kλi(u − ui)kT + kλuuik

Xi∈∂T Xi∈∂T ∩∂Ω X X ≤ ku − uikT + kuik

Xi∈∂T Xi∈∂T ∩∂Ω ! X X ≤ diam(Ω ) k∇uk + k∇uk , Xi ΩXi ΩXi Xi∈∂T Xi∈∂T ∩∂Ω where we used (2.73) and (2.74) in the end. Observing that both sums extend over ﬁnitely many vertices and that by local quasi-uniformity we can bound the diameter of diam(ΩXi ) by that of T , namely diam(ΩXi ) ≤ chT , we obtain the estimate in L2(T ). For the estimate in H1(T ), we observe that the mean value 1 calculation is a continuous operation on H0 (Ω(Xi)), and that (2.75) as a ﬁnite sum is continuous, therefore,

|u − Ihu|1;T ≤ k∇ukT + k∇IhukT ≤ ck∇ukΩ(T ). (2.76)

Finally, we use the trace estimate for u ∈ H1(T )

−1/2 1/2 kukF ≤ c h kukT + h k∇ukT , (2.77) to obtain the estimate on the edge.

2.3.6 Theorem (Scott-Zhang quasi-interpolation): Let {Th} be a locally quasi-uniform family of meshes with piecewise polynomial ﬁ- 1 nite element spaces Vh ⊂ H (Ω). Then, there exist bounded operators 1 1 Ih : H (Ω) → Vh such that for every function u ∈ H (Ω), every mesh cell T , every face F , and for m = 0, 1 there holds

1−m ku − Ihukm;T ≤ chT |u|1;ΩT (2.78) 1/2 ku − Ihuk0;F ≤ chT |u|1;ΩT , (2.79)

and Ihu is a quasi-interpolation on the boundary ∂Ω.

52 2.3.7 Theorem (Schöberl quasi-interpolation): Let {Th} be a locally quasi-uniform family of meshes with piecewise polynomial ﬁnite 1 element spaces Vh ⊂ H (Ω). Then, there exist bounded projection op- 1 1 erators Ih : H (Ω) → Vh such that for every function u ∈ H (Ω), every mesh cell T , every face F , and for m = 0, 1 there holds

1−m ku − Ihukm;T ≤ chT |u|1;ΩT (2.80) 1/2 ku − Ihuk0;F ≤ chT |u|1;ΩT . (2.81)

2.3.8 Deﬁnition: Let a(u, v) = f(v) be the weak formulation of a BVP on the space V . Then, we deﬁne the residual

R: V → V ∗ (2.82) w 7→ f(·) − a(w, ·).

1 2.3.9 Lemma: For u ∈ V = H0 (Ω) solution to Poisson’s equation and any other function w ∈ V , there holds

hRv, wi |u − v|1 = kRvk−1 := sup . (2.83) w∈V |w|1

Proof. We have

hRv, wi = f(w) − a(v, w) = a(u − v, w). (2.84)

Since a(., .) is s.p.d., we can apply Cauchy-Schwarz to obtain the result.

2.3.10 Deﬁnition: Let u be a piecewise continuous function on a mesh T. On a face F between two cells T1 and T2, we deﬁne the mean value operator

u (x) + u (x) u(x − εn ) + u(x − εn ) {{u}}(x) = 1 2 = lim 1 2 , (2.85) 2 ε&0 2

where ni are the outer normal vectors of the two cells. The jump operator is (u − u )n (u − u )n {{un}} = 1 2 1 = 2 1 2 . (2.86) 2 2

53 1 2.3.11 Deﬁnition: Let u ∈ V = H0 (Ω) be the solution to Poisson’s equation and let uh ∈ Vh be a ﬁnite element function. The strong form of the residual is X Z X hRv, wi = rT (uh)w dx − 2{{n · ∇uh}}w ds, (2.87) T ∈ T i Th F ∈Fh where

rT (uh) = f + ∆uh. (2.88)

2.3.12 Lemma: Let Th be a locally quasi-uniform mesh. There is a constant c > 0 such that the error between the true solution u ∈ V = 1 H0 (Ω) and the ﬁnite element solution uh ∈ Vh ⊂ V is bounded by

 1/2 X 2 2 X 2 |u − uh|1;Th ≤ c  hT krT (uh)kT + hF k{{n · ∇uh}}kF  . T ∈ i Th F ∈Fh (2.89)

Proof. We begin with the equivalence of error in V and residual in V ∗. For the residual, there holds Galerkin orthogonality, such that we obtain

hRuh, wi Ruh, w − Ihw |u − uh|1 = sup = sup . (2.90) w∈V |w|1 w∈V |w|1

Using the strong form and the quasi-interpolation Ih, we get

X Z X hRuh, wi = rT (uh)(w − Ihw) dx − 2{{n · ∇uh}}(w − Ihw) ds T ∈ T i Th F ∈Fh X X ≤ krT (uh)kT kw − IhwkT + k{{n · ∇uh}}kT kw − IhwkT T ∈ i Th F ∈Fh X X 1/2 ≤ c1 hT krT (uh)kT k∇wkΩT + c2 hE k{{n · ∇uh}}kT k∇wkΩT T ∈ i Th F ∈Fh Applying Hölder inequality, we obtain for the ﬁrst term

!1/2 !1/2 X X X h kr (u )k k∇wk ≤ h2 kr (u )k2 k∇wk2 , T T h T ΩT T T h T ΩT T ∈Th T ∈Th T ∈Th

54 and similar for the second term. Both contain a term of the form

X 2 X X 2 2 k∇wk = k∇wk 0 ≤ nk∇wk , ΩT T Ω 0 T ∈Th T ∈Th T ∈TT where n is the maximal number of occurrences of a cell T 0 in the double sum. This n is bounded uniformly on shape regular family of meshes, since shape regularity prohibits degeneration of cells. Therefore, we conclude

 1/2 X 2 2 X 2 hRuh, wi ≤ c  hT krT (uh)kT + hEk{{n · ∇uh}}kT  |w|1;Ω. T ∈ i Th F ∈Fh

Entering into (2.90) yields the proposition. Remark 2.3.13. The constant c in the previous theorem depends on the constant in the quasi-interpolation estimate, which in turn was derived using Poincaré and trace inequalities. Traditionally, both are derived using indirect arguments, but a constructive proof with computable bounds is possible. For details, see [Verfürth, 2013, Chapter 3]. There still remains the question, whether these bounds are suﬃciently sharp to be applicable in practice.

2.3.14 Deﬁnition: For a function f ∈ L2(Ω), we deﬁne the cell-wise average 1 Z f(x) = f dx x ∈ T,T ∈ Th, (2.91) |T | T and the data oscillation

oscT f = kf − fkT . (2.92)

2.3.15 Definition: The residual based error estimator for the finite element method for Poisson’s equation on a mesh T is defined as X ηh(uh) = ηT (uh), (2.93) T ∈T where 1 X η (u )2 = h2 kf + ∆u k2 + h k{{n · ∇u }}k2 . (2.94) T h T h T 2 F h E F ⊂∂T

55 2.3.16 Theorem: There holds with positive constants c1 and c2

2 2 X 2 2 c1|u − uh|1 ≤ ηh(uh) + hT oscT f (2.95) T ∈Th X c η2 (u ) ≤ |u − u |2 + h2 osc2 f (2.96) 2 T h h 1;ΩT T T T ∈TT

Proof. Note that the right hand side of the estimate (2.89) in Lemma 2.3.12 diﬀers from the estimator ηh(uh) only by the replacement of f by f. Therefore, we can start the proof of this lemma by changing (2.90) to hRuh, wi Ruh, w + f − f, w |u − uh|1 = sup = sup , (2.97) w∈V |w|1 w∈V |w|1 where we ad hoc deﬁned R like R, but replacing rT = f + ∇uh by f + ∇uh. Due to equality, we can still subtract the quasi-interpolant of w and continue through the whole proof. What remains is the estimate

!1/2 X X f − f, w − Ihw ≤ c hT kf − fkT k∇wkΩT ≤ c hT kf − fkT |w|1;Ω, T ∈T T ∈T (2.98) which is proven by Hölder inequality and the boundedness of the number of cells in TT . Thus, the upper bound (2.95) is an immediate consequence of Lemma 2.3.12. Note: the proof for the lower bound is for linear elements only. It can be generalized to higher order by generalizing the bubble functions below, which is mostly technical. We observe that the lower bound is local, that is, the estimate on each cell T is bounded from above by the error on the halo ΩT . Therefore, we start by constructing test functions with support on a cell. On a simplex, we deﬁne the bubble function in terms of barycentric coordinates 1 B = λ λ . . . λ , (2.99) T (d + 1)d+1 0 1 d while on the reference hypercube (0, 1)d we let

BbT (xb) = b(xb1)b(xb2) . . . b(xbd) b(x) = 4x(1 − x). (2.100) Both share the following properties: they vanish on ∂T , they are positive inside T , and max BT = 1. From positivity, we deduce the existence of a constant c depending on shape regularity and the shape function space P(T ), such that Z 2 2 p BT dx ≥ ckpkT ∀p ∈ P(T ). (2.101) T

56 Furthermore, BT is polynomial, such that the inverse estimate holds.

Choose now the test function wT = (f + ∆uh)BT . Since it vanishes on the boundary and outside of T , the strong and weak form of the residual reduce to Z Z rT (uh)wT dx = ∇(u − uh) · ∇wT dx. (2.102) T T

Adding the diﬀerence of f and f on both sides yields Z Z Z 2 (f + ∆uh) BT dx = ∇(u − uh) · ∇wT dx + (f − f)wT dx. (2.103) T T T On the left, we estimate Z 2 2 ckf + ∆uhkT ≤ (f + ∆uh) BT dx. (2.104) T On the right, we estimate the residual term by Z ∇(u − uh) · ∇wT dx ≤ |u − uh|1;T |wT |1;T T −1 ≤ c|u − uh|1;T hT kf + ∆uhkT . The data oscillation is estimated in a straightforward way by Z (f − f)wT dx ≤ oscT fkf + ∆uhkT T Combining these three estimates yields

−1 ckf + ∆uhkT ≤ hT |u − uh|1;T + oscT f, (2.105) which is the desired estimate for the cell term of the estimator ηT (uh) if we multiply by hT . The estimate for the face term is similar, using a bubble function for the face. Since this functon is nonzero on the face, its support extends over two mesh cells. On simplicial cells, it is constructed like the cell bubble in equation (2.99), but extending the product only over indices of vertices on the face. Therefore, it is a polynomial of degree d − 1 on the face and also in the interior of each cell. For hypercubes, it is the product of the quadratic polynomials b(x) of variables on the face times a linear function decaying linearly from 1 to zero. Again, we 1 normalize such that the maximum equals 1. Choosing wF = 2{{n · ∇uh}}F BF ,

1For linear elements, the derivative on the face is constant and thus this deﬁnition is obvious. For higher order polynomials, we need a an extension from the face into the interior, which can be obtained from the barycentric coordinates or the tensor product structure.

57 we obtain Z 2 2 ck{{n · ∇uh}}kF ≤ {{n · ∇uh}} BF ds F Z = {{n · ∇uh}}wF ds F X Z = rT wF dx − hR, wF i T T ∈TF X Z Z = rT wF dx − ∇(u − uh) · ∇wF dx T ΩF T ∈TF X Z Z X Z = (f + ∆uh)wF dx − ∇(u − uh) · ∇wF dx + (f − f)wF dx. T ΩF T T ∈TF T ∈TF Again, we estimate the three terms on the right hand side separately. For the ﬁrst, we estimate the norm of wF by its trace on the edge F . From the already proven estimate for the cell residual (2.105), we obtain

X Z X (f + ∆uh)wF dx ≤ kf + ∆uhkT kwF kT T T ∈TF T ∈TF X 1/2 ≤ kf + ∆uhkT chF k{{n · ∇uh}}kF T ∈TF X −1/2 1/2 ≤ +c hF |u − uh|1;T + hF oscT f k{{n · ∇uh}}kF T ∈TF

Similarly, the inverse estimate for wF yields Z

∇(u − uh) · ∇wF dx ≤ |u − uh|1;ΩF k∇wF kΩF ΩF −1/2 ≤ |u − uh|1;ΩF chF k{{n · ∇uh}}kF . Finally, the data oscillation terms becomes

X Z X (f − f)wF dx ≤ kf − fkT kwF kT T T ∈TF T ∈TF X 1/2 ≤ kf − fkT chF k{{n · ∇uh}}kF . T ∈TF

1/2 Summing and scaling with hF yields the result.

58 Chapter 3

Variational Crimes

3.0.1. In the previous chapter, we considered ﬁnite element methods applying the original bilinear form to a subspace Vh ⊂ V . This assumes, that the domain Ω is exactly represented by the mesh, and that all integrals are computed exactly. Both assumptions reduce the applicability of the ﬁnite element method considerably. Therefore, we now extend our analysis to cases, where we allow Vh 6⊂ V and discrete bilinear forms ah(., .) 6= a(., .).

3.1 Numerical quadrature

3.1.1. We begin the investigation of variational crimes by studying the eﬀect of using numerical quadrature instead of exact integration on mesh cells. In particular, we investigate approximations of the form

nq Z X f(x) dx ≈ ωkf(xk) =: QT (f). (3.1) T k=1

1 First, we observe that QT is not a bounded operator on L (Ω), such that QT (∇u· ∇v) is undeﬁned for functions in H1(Ω). The surprising result of this section is, that quadrature is still admissible for the implementation of a ﬁnite element method.

We will ﬁrst set a theoretical framework and then investigate quadrature rules in detail. The presentation follows [Ciarlet, 1978, Chapter 4].

59 3.1.2 Lemma (Strang’s ﬁrst lemma): Let a(., .) be a bounded and elliptic bilinear form on the Hilbert space V . Let Vn ⊂ V and let ah(., .) be a bilinear form, bounded and elliptic on Vn with constants Mn and ∗ αn. Let f, fn ∈ V . If u ∈ V and un ∈ Vn ⊂ V are solutions to

a(u, v) = f(v) ∀v ∈ V,

an(un, vn) = fn(vn) ∀vn ∈ Vn,

respectively, there holds

ku − unkV M 1 ≤ inf 1 + ku − v k + ka (v ,.) − a(v ,.)k ∗ n V n n n Vh vn∈Vn αn αn 1 + kf − fk ∗ . (3.2) n Vh αn

Proof. Using Vn-ellipticity of an(., .) yields for arbitrary vn ∈ Vn

2 αnkun − vnk ≤an(un − vn, un − vn)

=fn(un − vn) + a(u − vn, un − vn) − f(un − vn)

+ a(vn, un − vn) − an(vn, un − vn)

We estimate separately

ku − unk ≤ ku − vnk + kun − vnk 1 ≤ ku − v k + Mku − v k + ka (v ,.) − a(v ,.)k ∗ + kf − fk ∗ . n n n n n Vh n Vh αn (3.3)

Remark 3.1.3. Strang’s lemma states, that the error can be split into the approximation error of the space Vh and the consistency error of the approxi- 1 mations ah(., .) und fh(.). Both errors are scaled with the stability factor /αn

60 of the discrete problem. So far, this is consistent with error estimates for instance for Runge-Kutta methods. There is an important diﬀerence though: the consistency error is not evaluated for the exact solution, but only for its discrete approximation. Remark 3.1.4. We will apply Strang’s lemma to a family of meshes indexed by mesh size h and assess the inﬁmum by an interpolation operator. It is clear, that we will only obtain optimal convergence rates compared to the interpolation estimate, if there exists α0 > 0 such that αn ≥ α0 uniformly with respect to n. While it is not a prerequisite of Strang’s lemma, it is our goal for all discretizations. Remark 3.1.5. Quadrature would be infeasible, if we had to devise a quadrature rule for every mesh cell T . Instead, we tabulate quadrature formulas for the reference cell Tb by choosing quadrature points xbk and weights ωk and write

nq X Q (f) = ω f(x ). (3.4) Tb b k b bk k=1 We compute integrals over T though mapping,

nq X QT (f) = det ∇ΦT (xbk)ωkfb(xbk). (3.5) k=1

Thus, QT is deﬁned by quadrature points xk = Φ(xbk) and quadrature weights det ∇ΦT (xbk)ωk. Quadrature rules on the reference cell Tb are obtained by interpolation after choosing quadrature points by employing the properties of orthogonal polynomials. The construction of such quadrature rules for simplices is somewhat complicated, and we refer to tables in the cited literature. An important consequence of the use of quadrature points as roots of orthogonal polynomials is the fact that all weights are positive.

3.1.6 Deﬁnition: Given a one-dimensional quadrature rule QI on the interval I = [0, 1] with

nq X QI (fb) = ωkfb(xbk). (3.6) k=1

Then, a quadrature rule on Tb = [0, 1]d is deﬁned by

nq nq X X Q (f) = ··· ω ··· ω f(x , . . . , x ). (3.7) Tb b k1 kd b k1 kd k1 kd

61 3.1.7 Lemma: If QI is an n-point Gauß formula, the tensor product quadrature in the preceding deﬁnition is exact on Q2n−1.

Remark 3.1.8. When we look at Poisson’s equation on simplicial meshes, the integrals Z Z −T −T aT (ϕj, ϕi) = ∇ϕj · ∇ϕi dx = ∇Φ ∇b ϕbj · ∇Φ ϕbi det ∇Φ dxb, (3.8) T Tb can be computed exactly, since ∇Φ is a constant matrix. Already on arbitrary quadrialterals, ∇Φ−T is a rational function, wich is harder to integrate. But, in order to justify the use and investigate the properties of approximations by numerical quadrature, we are considering a model problem with varying coeﬃcients.

3.1.9 Assumption: Let in this section the bilinear form be deﬁned as Z a(u, v) = ∇uA∇vT dx, (3.9) Ω where A = A(x) is a matrix with

T 2 d ξ A(x)ξ ≥ α|ξ | ∀ξ ∈ R , ∀x ∈ Ω. (3.10)

1 The function u ∈ V = H0 (Ω) is the solution to

a(u, v) = (f, v) ∀v ∈ V. (3.11)

We assume that the coeﬃcients aij and the right hand side f are functions which are well-deﬁned in all quadrature points on all meshes of a family {Th}.

3.1.10 Deﬁnition: The bilinear form ah(., .) obtained by numerical quadrature Q on the mesh is deﬁned as Tb Th

X T ah(u, v) = QT (∇uA∇v ), (3.12) T ∈Th where Q is obtained from Q by the mapping Φ . The right hand side T Tb T is X fh(v) = QT (fv). (3.13) T ∈Th

62 3.1.11 Lemma: Let a ﬁnite element method be deﬁned by a mesh and a shape function space on the reference cell T . Let Q Th P b Tb be a quadrature formula, such that all quadrature weights are positive. Furthermore, assume that one of the following holds:

1. The function values in the quadrature points xbk are unisolvent on G1 = ∂ip p ∈ P, i = 1, . . . , d , (3.14)

2. The quadrature rule is exact on G2 = ∇p · ∇q p, q ∈ P . (3.15)

Then, there is a constant α0 > 0 depending on shape regularity and the quadrature rule, but independent of h, such that

2 ah(uh, uh) ≥ α0|uh|1;Ω ∀uh ∈ Vh. (3.16)

Remark 3.1.12. If P = Pk, then G1 = Pk−1 and G2 = P2k−2. For Qk, these relations are more complicated, and the simplest we can say is G1 ⊂ Qk and G2 ⊂ Q2k

Proof of Lemma 3.1.11. Assume ﬁrst condition 1. From the poisitivity of quadrature weights, we conclude that

nq d X X Q (∇p · ∇p) = 0 = ω (∂ p(x ))2 = 0 (3.17) Tb k i bk k=1 i=1 implies ∂ p(x ) = 0. Unisolvence implies ∂ p ≡ 0. Therefore, pQ (∇p · ∇p) i bk i Tb defines a norm on the space P/R. Since this space is finite dimensional and |.| is a second norm, we conclude from norm equivalence the existence of a 1;Tb constant c > 0 such that c|p|2 ≤ Q (∇p · ∇p) ∀p ∈ P. (3.18) 1;Tb Tb On the other hand, condition 2 yields the same estimate with c = 1. It is easily verified, that |p|2 and Q (∇p·∇p) scale equally under the mapping 1;Tb Tb ΦT , such that the same estimate holds on T with a different constant depending on shape regularity, again denoted by c. Finally, 2 X 2 α0c|uh|1;Ω = α0c |uh|1;T T ∈Th X ≤ α Q (∇u · ∇u ) 0 Tb h h T ∈Th X ≤ Q (∇u A∇uT ) = a (u , u ). Tb h h h h h T ∈Th

63 3.1.13. Now that we have established conditions for stability of the discrete problem, we can address an estimate of the consistency error. In order to avoid additional overhead in an already technical argument, we restrict the analysis to the spaces PT = Pk on simplices. Instead of proving a general result for arbitrary quadrature formulas, we ﬁrst take ask, what the order of the quadrature error should be. Indeed, we have the interpolation estimate k ku − Ihuk1 = O(h ). Therefore, we investigate quadrature rules introducing consistency errors of the same order.

3.1.14 Theorem: Assume in addition to Assumption 3.1.9 for k ≥ 1 k,∞ that for all cells PT = Pk and aij ∈ W (Ω). Let the quadrature rule Q be exact for polynomials in . Then, there exists a constant c Tb P2k−2 such that for all meshes of a quasi-uniform family {Th} of meshes there holds

k |a(vh, wh) − ah(vh, wh)| ≤ ch kAkk,∞;Ωkvhkk;h|wh|1;h. (3.19)

3.1.15 Lemma: Let u ∈ W k,p(Ω) and v ∈ W k,∞(Ω). Then, uv ∈ W k,p(Ω) and

k X |uv|k,p;Ω ≤ c |u|i,q;Ω |v|k−i,∞;Ω, (3.20) i=0 with a constant c depending on k and the space dimension, but not on Ω.

Proof. This is basically the product rule. For a multi-index α with |α| = k, we have

k X X ∂α(uv) = ∂βu ∂α−βv. i=0 |β|=i βj ≤αj

Then, Hörder inequality is applied to the sums and the integrals.

Proof of Theorem 3.1.14. We ﬁrst argue on the reference cell and deﬁne the quadrature error Z E (f) = f dx − Q (f). (3.21) Tb b Tb Tb

64 Our goal is the estimation of E (apq) for functions a ∈ W k,∞(T ) and poly- Tb b nomials p, q ∈ Pk−1, such that we obtain an estimate for the constituents of E (∇uA∇v). Tb b b bb b We combine ϕ = ap ∈ W k,∞(Tb) Since W k,∞(Tb) ,→ C(Tb), we obtain |E (ϕq)| ≤ ckϕqk ≤ ckϕk kqk ≤ ckϕk kqk . (3.22) Tb ∞ 0,∞ 0,∞ k,∞ L2(Tb) For the last inequality, we used the natural bound of the norm in W 0,∞ by that of W k,∞ and the norm equivalence on . We conclude that J (ϕ) = E (ϕq) Pk−1 q Tb k,∞ is a bounded linear functional on W . Furthermore, it vanishes for ϕ ∈ Pk−1 by the assumption on the quadrature rule. Thus by the Bramble-Hilbert lemma (Lemma 2.2.3), |E (ϕq)| ≤ c|ϕ| kwk . (3.23) Tb k,∞ L2(Tb) Now, we apply Lemma 3.1.15, to obtain

|ϕ|k,∞ = |ap|k,∞ k−1 X ≤ c |a|k−i,∞|p|i,∞ i=0 (3.24) k−1 X ≤ c |a|k−i,∞|p|i, i=0 where we used that |p|k = 0 and norm equivalence on Pk−1. Collecting every- thing and reintroducing ba, pb, and qb for a, p, and q, respectively, we obtain k−1 ! X |E (apq)| ≤ c |a| |p| kqk . (3.25) Tb bbb b k−1,∞;Tb b i;Tb b L2(Tb) i=1 The scaling lemma yields |a| ≤ chk−i|a| 0 ≤ i ≤ k − 1, (3.26) b k−1,∞;Tb T k−i,∞;T d |p| ≤ chi− /2|p| 0 ≤ i ≤ k − 1, (3.27) b i;Tb T i;T −d/2 kqk ≤ ch kqk 2 . (3.28) b L2(Tb) T L (T ) Therefore,

d |E (apq)| ≤ ch /2|E (apq)| T T Tb bbb k−1 ! k X ≤ chT |a|k−i,∞;T |p|i;T kqkL2(T ) (3.29) i=1 k ≤ chT kakk−i,∞;T kpki;T kqkL2(T ).

Entering the derivatives of uh and vh for p and q, respectively, and summing up over all cells yields the result.

65 Remark 3.1.16. Note that the assumption of quasi-uniformity is convenient for notation, but excessive. Indeed, shape regularity is suﬃccient, but in that case, the estimate must be localized, that is,

X k a(vh, wh) − ah(vh, wh) ≤ c hT kAkk,∞;T kvhkk;T |wh|1;T . (3.30) T ∈Th

3.1.17 Theorem: Let the ﬁnite element use shape function spaces = and let the quadrature rule Q be exact for polynomials PT Pk Tb in P2k−2. Then, there exists a constant c such that for all cells of a quasi-uniform family {Th} of meshes there holds

k k |ET (fp)| ≤ chT kfkk;T kpk1;T ∀f ∈ H (Ω), ∀p ∈ P. (3.31)

66 Chapter 4

Solving the Discrete Problem

4.0.1. The motivation for the use of iterative methods lies in the fact that matrices resulting from the discretization of partial diﬀerential equations tend to be very big, but sparse, that is, most of their entries are zero. Let us take for example trilinear ﬁnite elements on a uniform grid of (n − 1) cells in each direction. This leeds to n3 degrees of freedom which we number lexicographically. The matrix stencil, that is, the distribution of nonzero entries, in one dimension is tridiagonal, that is, ∗ ∗  ∗ ∗ ∗  S =   1  ......   . . . ∗ ∗ In two dimensions, we obtain the stencil   S1 S1 S1 S1 S1  S = S ⊗ S =   , 2 1 1  ......   . . . S1 S1 and in three dimensions   S2 S2 S2 S2 S2  S = S ⊗ S ⊗ S =   . 3 1 1 1  ......   . . . S2 S2

The corresponding matrix has N = n3 rows and columns. Let us compare some solution methods for n = 100 with respect to memory and a hypothetic hardware which executes 1010 multiplications per second (additions are free).

67 1 3 2 Gaussian elimination The effort needed is 3 N +O(N ), leading to the computing time 1018 T = sec ≈ 3 · 107 sec ≈ 1.06 years G 3 · 1010 The backward substitution is only of order N 2 and can be neglected. The memory requirement with double precision is 8 · 1012 Bytes, almost 10 terabytes. Banded LU decomposition Here we make use of the fact that LU decomposition can be restricted to the hull of the outermost nonzero elements of the matrix, the so called banded or skyline version. Let M be the great- est distance of a nonzero matrix element aij from the diagonal, that is, 1 2 M = |i − j|. Then, the leading term of the effort is 3 N ·M , yielding with M = n2 a computing time of 106 · 104 · 104 T = sec ≈ 3 · 103 sec ≈ 50 minutes BLU 3 · 1010 The storage requirement for N · M double precision numbers is 8 · 1010 Bytes, almost 100 gigabytes. matrix vector product For comparison, the multiplication with such a matrix, given that there are at most 9 nonzero entries per row costs 9 · 106 T = ≈ 1 msec, mult 1010 that is, we can perform more than 106 matrix-vector multiplications before we reach the effort of the banded LU decomposition. The storage requirement is roughly 1 gigabyte and can be reduced to almost zero by a smart implementation. Remark 4.0.2. For purposes of analysis we typically choose the space X = L2(Ω). We admit a small inaccuracy here: when we run the algorithms on a computer, we usually employ the Euclidean inner product, thus X should be the space of degrees of freedom. But this is a discrete space, where we cannot use theory of function spaces easily. Instead, we note that the L2-inner product of standard finite element bases yield inner products equivalent to the Euclidean up to the local mesh size (see Lemma ??). Example 4.0.3. While the methods developed in this chapter are fairly general, we introduce a specific model problem as a simple benchmark case. To this end, 1 we consider the Dirichlet problem: find u ∈ V = H0 (Ω) such that Z Z a(u, v) ≡ ∇u · ∇v dx = fv dx ≡ f(v), ∀v ∈ V. (4.1) Ω Ω The finite dimensional linear systems of equations are derived from finite element discretizations on quasi-uniform meshes of cells with maximal diameter h,

68 yielding a sequence of spaces Vh, on which linear systems are introduced by the same weak form (4.1). Notation 4.0.4. With a bilinear form a(., .) on X×X we associate the operator A : X → X by

hAu, vi = a(u, v), ∀v ∈ V, (4.2) where now V = D(A) is the domain of A, that is, the subset of functions v ∈ X, such that Av is defined and in X. We will tacitly assume that operators A, B, etc. are defined by equation (4.2) and the bilinear forms a(., .), b(., .), etc., respectively, if they are not defined otherwise.

4.0.5 Deﬁnition: We call the bilinear form a(., .) and its associated operator A symmetric, if there holds

a(u, v) = a(v, u) ∀u, v ∈ V.

They are called V -elliptic, if for there is a positive number γ such that

2 a(u, u) ≥ γkukV ∀u ∈ V.

4.0.6 Definition: For positive definite, symmetric operators, we obtain the possibly infinite bounds of the spectrum

a(u, u) a(u, u) Λ(A) = sup 2 , λ(A) = inf 2 , (4.3) u∈V kukX u∈V kukX as well as the possibly inﬁnite spectral condition number

Λ(A) κ(A) = . λ(A)

Remark 4.0.7. Note that the spectral condition number depends on the norm of the space X. It is bounded, if and only if A is bounded with respect to this norm.

1 Example 4.0.8. Let X = H0 (Ω) with the inner product Z hu, vi1 = ∇u · ∇v dx. Ω If A is the operator associated with the bilinear form a(., .) in (4.1), then

Λ(A) = λ(A) = κ(A) = 1.

69 If on the other hand X = L2(Ω) equipped with the usual L2-inner product, then A is unbounded and thus κ(A) = ∞. λ(A) is the constant in Friedrichs’ inequality.

Notation 4.0.9. After choosing a basis for a finite dimensional space Xn or a Schauder basis for the space X (assuming X separable), say {ϕi}, we can define a (possibly infinite-dimensional) matrix A associated with the bilinear form a(., .) with the entries

aij = a(ϕj, ϕi).

If we restrict the bilinear forms to a ﬁnite dimensional subspace Xn, we denote the matrices A restricted to this subspace by An.

4.0.10 Deﬁnition: The two extremal eigenvalues of the matrix An can be obtained by the maximum and minimum of the Rayleigh quotient

xT Ax xT Ax Λ(A) = max , λ(A) = min . (4.4) n T n T x∈R x x x∈R x x The spectral condition number is

Λ(A) κ (A) = . n λ(A)

Remark 4.0.11. The spectral condition number of the operator A depends on the bilinear form a(., .) and the choice of the norm in X. On the other hand, the spectral condition number of the matrix A depends on the choice of a basis of the space Xn.

4.1 The Richardson iteration

4.1.1. As a ﬁrst example and prototype for all other iterative methods we consider Richardson’s method, which for matrices and vectors in Rn reads

(k+1) (k) (k) x = x − ωk Ax − b . (4.5)

ωk is a relaxation parameter, which can be chosen a priori or can be changed in every step. We will for simplicity assume ωk = ω.

70 4.1.2 Lemma: The error after one step of the Richardson method is given by x(k+1) − x = E x(k) − x , (4.6)

where the error propagation operator is

E = I − ωA. (4.7)

Proof. Using the fact that x = A−1b, we write

x(k+1) − x = x(k) − ωAx(k) − b − x(k) = x(k) − x − ωAx(k) − x.

4.1.3 Theorem: If A is symmetric, positive deﬁnite, with extremal eigenvalues λ > 0 and Λ > 0, then Richardson’s method converges if and only if 0 < ω < 2/Λ. The optimal relaxation parameter is 2 ω = , (4.8) opt λ + Λ which yields an optimal contraction rate of 2λ Λ − λ κ − 1 2 % = 1 − = = = 1 − + O κ−2 , (4.9) opt λ + Λ Λ + λ κ + 1 κ where κ = Λ/λ is the so called spectral condition number.

Proof. Convergence of this method is analyzed through the Banach ﬁxed-point theorem, which requires contraction property of the matrix M = I − ωA. Al- ternatively, we studied a theorem that states, that a matrix iteration converges if and only if the spectral radius

%(M) = max |λ(M)| < 1, the maximum absolute value of the eigenvalues of M is strictly less than one.

If A is symmetric, positive deﬁnite, with eigenvalues λi > 0, we have that

%(M) = max |1 − ωλi| . (4.10) i

71 Let the extremal eigenvalues be determined by the minimum and maximum of the Rayleigh quotient, xT Ax xT Ax λ = min , and Λ = max . (4.11) n T n T x∈R x x x∈R x x Then, equation (4.10) yields that the method converges for 0 < ω < 2/Λ. Furthermore, for 1/Λ ≤ ω ≤ 2/Λ we have %(M) = max−1 + ωΛ, 1 − ωλ . The optimal parameter ω is the one where both values are equal and thus (4.8) and (4.9) hold. 4.1.4. The analysis of finite element methods shows that it is beneficial to give up the focus on finite dimensional spaces and rather use theory that applies to separable Hilbert spaces. If results can obtained in this context, they can easily be restricted to finite dimensional subspaces and thus become uniform with respect to the mesh parameter. Thus, we will first reformulate Richardson’s method for this case and then derive convergence estimates. 4.1.5. Elements of an abstract Hilbert space X will be denoted by u, v, w, etc. On the other hand, coefficient vectors in Rn are denoted by letters x, y, z, etc.

4.1.6 Deﬁnition: Let X be a Hilbert space with inner product h., .i. Let a(., .) be a second bilinear form on X and the domain of its operator is V . Then, for any right hand side f ∈ V and any start vector u(0) ∈ V , Richardson’s method is deﬁned by the iteration

D (k+1) E D (k) E (k) u , v = u , v − ωk a(u , v) − hf, vi , ∀v ∈ X. (4.12)

ωk is a suitable relaxation parameter, chosen such that the method converges.

4.1.7. The scalar products in (4.12) become necessary, since diﬀerent from the case in Rn, the result of applying the bilinear form a(., .) to u(k) in the ﬁrst argument yields a linear form on X. In order to convert this to a vector in X, we have to apply the isomorphism induced by the Riesz representation theorem.

4.1.8 Theorem: Let the bilinear form a(., .) be bounded and elliptic on X × X, namely, let there exist positive constants Λ and λ such that for all u, v ∈ X there holds

a(u, v) ≤ Λkukkvk, a(u, u) ≥ λkuk2. (4.13)

Then, Richardson’s iteration converges for ωk = ω for any ω ∈ (0, 2λ/Λ2).

72 Proof. We deﬁne the iteration operator T as the solution operator of equation (4.12), namely T u(k) := u(k+1). We have to prove that T is a contraction on X under the assumptions of the theorem.

For two arbitrary vectors u1, u2 ∈ X, let w = u1 − u2 be their diﬀerence. Due to linearity, we have T w = T u1 − T u2 and

hT w, vi = hw, vi − ωa(w, v) = hw − ωAw, vi.

Using v = T w as a test function, we obtain

kT wk2 = hw − ωAw, w − ωAwi = kwk2 − 2ωa(w, w) + ω2kAwk2 ≤ kwk2 − 2λωkwk2 + Λ2ω2kwk2 = 1 − 2λω + Λ2ω2kwk2. | {z } =:%(ω)

The function %(ω) is a parabola open to the top, which at zero equals one and has a negative derivative. Thus, it is less than one for small positive valuers of ω. The other point where %(ω) = 1 is ω = 2λ/Λ2.

Remark 4.1.9. The condition on ω in Theorem 4.1.8 is more restrictive than in Theorem 4.1.3, since λ/Λ ≤ 1. This is due to the fact, that in Theorem 4.1.3 we assume symmetry, and thus orthogonal diagonalizability of the matrix A. With similar assumptions, Theorem 4.1.8 could be made sharper. Remark 4.1.10. It is clear that the boundedness and ellipticity estimates (4.13) hold for any ﬁnite dimensional subspace Xn ⊂ X, and thus the convergence estimate (4.9) becomes independent of n.

More interesting and also more common is the case where the bilinear form a(., .) is unbounded on X. While it is still bounded on each ﬁnite subspace Xn, this bound cannot be independent of n if the sequence {Xn} approximates X.

4.1.11 Deﬁnition: We deﬁne an operator B : X → X∗ such that Bu = b(u, .) := hu, .i. By the Riesz representation theorem, there is a continuous inverse operator B−1 : X∗ → X, which is often called Riesz isomorphism.

73 4.1.12 Deﬁnition: When we apply Richardson’s method as in (4.12) on a computer, each step involves a multiplication with the matrix A, but an inversion of the matrix B, corresponding to the iteration

(k+1) (k) (k) Bx = Bx − ωk Ax − b ,

or equivalently,

(k+1) (k) −1 (k) x = x − ωkB Ax − b . (4.14)

Remark 4.1.13. The iteration in (4.14) is commonly referred to as preconditioned Richardson iteration and B−1 as the preconditioner. Note that by introducing the iteration in its weak form (4.12), the preconditioner arrives naturally and with necessity. The goal of this chapter is ﬁnding preconditioners B−1, or equivalently inner products h., .i, such that the bilinear form a(., .) is bounded and the condition number κ = Λ/λ is small. In order to reduce (or increase) confusion, we will refer to the inner product that we search in order to bound the condition number as b(., .) instead of h., .i, this way separating the Hilbert space X more clearly from the task of preconditioning. Thus, the operator B and the matrix B will be associated with a bilinear form b(., .) and the ﬁnal version of the preconditioned Richardson iteration is

(k+1) (k) (k) b(u , v) = b(u , v) − ωk a(u , v) − f(v) , ∀v ∈ X, (4.15) or in operator form

(k+1) (k) −1 (k) u = u − ωkB (Au − f). (4.16) Remark 4.1.14. The space X and is inner product does not appear anymore in this formulation, since the bilinear form b(., .) has replaced it. Thus, finding a preconditioner also amounds to changing the space in which we iterate. This is reflected by the following: Corollary 4.1.15. Let the symmetric bilinear forms a(., .) and b(., .) in the Richardson iteration (4.15) be both bounded and positive definite on the same space V and fulfill the spectral equivalence relation

λb(u, u) ≤ a(u, u) ≤ Λb(u, u), ∀u ∈ V. (4.17)

Then, if ωk ≡ ω ∈ (0, 2Λ), the iteration is a contraction on V . The optimal contraction number is % according to equation (4.9) for ω chosen as in (4.8).

Proof. This corollary is equivalent to Theorem 4.1.8 if the inner product h., .i is replaced by the bilinear form b(., .).

74 Remark 4.1.16. Originally, the space V was chosen as the domain of A which essentially meant V ⊂ H2(Ω), since we required Av ∈ X. An additional benefit of the preconditioned version is, that now V ⊂ H1(Ω) is sufficient and at least here no regularity assumption is required. Notation 4.1.17. In order to distinguish different preconditioners, we will also us the notation λ(B,A) and Λ(B,A) to refer to the constants in the norm equivalence (4.17). Example 4.1.18. Let us take the example (4.1). By the Poincaré-Friedrichs inequality, a(., .) is an inner product on X and thus we can choose h., .i = a(., .). In particular, λ = Λ = 1 and the optimal choice is ω = 1. Then, Richardson’s iteration becomes

a(u(k+1), v) = a(u(k), v) − a(u(k), v) − f(v) = f(v), ∀v ∈ X, which converges in a single step, but we have to solve the original equation for u. Thus, either the inversion of the matrix An is trivial on each ﬁnite dimensional subspace Xn, or the method is useless. With usual ﬁnite element bases, the latter is true.

Example 4.1.19. In the other extreme, we would like to use the Rn or L2 inner product on Xn or X, such that the Riesz isomorphism is easily computable. But then, the bilinear form a(., .) is unbounded on X. Thus, while for each ﬁnite n, the condition number κn = Λn/λn exists, it converges to inﬁnity if n → ∞.

4.2 The conjugate gradient method

4.2.1. Relying on Hilbert space structure more than Richardson’s iteration is the conjugate gradient method (cg), since it uses orthogonal search directions. Nevertheless, it also relies on constructing search directions from residuals, such that a Riesz isomorphism enters the same way as before and can then be used for preconditioning.

The beauty of the conjugate gradient method is, that it is parameter and tuning free, and it converges considerably faster than a linear iteration method.

75 4.2.2 Deﬁnition (Method of steepest descent): Let a(., .) be a symmetric, positive deﬁnite bilinear form on V . Then, the method of steepest descent for the energy functional

1 E(v) = 2 a(v, v) − f(v), (4.18) reads: given an initial vector u(0), compute for k ≥ 0 iteratively D E D E p(k), v = −∇E(u(k)), v ∀v ∈ V (4.19) V ∗×V (k) (k) αk = argmin E u + αp (4.20) α>0 (k+1) (k) (k) u = u + αkp (4.21)

4.2.3 Lemma: Let u ∈ V be the unique minimizer of the energy functional E(.). Then, there holds for all k a u(k+1) − u, p(k) = a(u(k+1), p(k)) − f(p(k)) = 0. (4.22)

With r(k) = f − a(u(k),.) ∈ V ∗, there holds

(k) (k) r , p ∗ α = V ×V . (4.23) k a p(k), p(k)

76 4.2.4 Deﬁnition: Let V be a Hilbert space and V ∗ its dual. The conjugate gradient method for an iteration vector u(k) ∈ V involves the residuals r(k) ∈ V ∗ as well as the update direction p(k) ∈ V and the auxiliary vector w(k) ∈ V . It consists of the steps 1. Initialization: for f and u(0) given, compute r(0) = f − a(u(0),.) and D E D E D E p(0), v = w(0), v = r(0), v ∀v ∈ V. V ∗×V

2. Iteration step: for u(k), r(k), w(k), and p(k) given, compute

(k) (k) r , w ∗ u(k+1) = u(k) + α p(k) α = V ×V k k a p(k), p(k)

(k+1) (k) (k) r = r − αka p ,. D E D E w(k+1), v = r(k+1), v ∀v ∈ V V ∗×V (k+1) (k+1) r , w ∗ p(k+1) = w(k+1) + β p(k) β = V ×V k k (k) (k) r , w V ∗×V

4.2.5 Definition: The preconditioned cg method is obtained from above algorithm by reinterpreting the Riesz isomorphism in the com- putation of w(k+1) as a preconditioning operation, much alike Defini- tion 4.1.12 of the preconditioned Richardson iteration. Thus, the line defining w(k+1) is replaced by

b(w(k+1), v) = r(k+1)(v) ∀v ∈ V.

Here, like there, the preconditioner enters naturally from the weak form of the algorithm.

4.2.6 Deﬁnition: The nth Krylov space as subspace of the Hilbert space V with inner product b(., .) of the operator A and seed vector w ∈ V is

−1 −1 −1 2 −1 n−1 Kn = Kn(B A, w) = span w, B Aw, (B A) w, . . . , (B Aw) . (4.24)

77 4.2.7 Lemma: The vectors generated by the conjugate gradient iteration have the following properties: either u(k) is the solution to the linear system or

r(k) = f − a(u(k),.) (4.25) D E D E r(k), w(k) = r(k), p(k) (4.26) V ∗×V V ∗×V D E r(k), p(j) = 0 j < k (4.27) a(p(k), p(j)) = 0 j < k (4.28) D E r(k), r(j) = 0 j < k (4.29)

4.2.8 Lemma: The iterates of the cg method have the following mini- mization properties:

(k) (0) ku − ukA = min ku + v − ukA v∈Kk (0) −1 (4.30) = min ku + p(B A)w − ukA. p∈Pn−1 p(0)=1

4.2.9 Theorem: Let the bilinear form a(., .) be symmetric, and let the spectral equivalence hold. Then, the preconditioned cg method converges and we have the estimate √ κ − 1k ku(k) − uk ≤ 2 √ ku(0) − uk . (4.31) A κ + 1 A

Here, κ = Λ/λ is the spectral condition number of the preconditioned problem.

78 4.3 Condition numbers of ﬁnite element matrices

4.3.1 Notation: In asymptotic estimates, we will use the notation

a . b and b & a, if there is a constant c independent of the relevant quantities, such that

a ≤ c b and c b ≥ a.

We use

a ' b,

if both hold. The latter is more precise than a = O(b), since it implies the same asymptotic order.

Example 4.3.2. The interpolation estimate (2.50) could be written as

k+1−m ku − Ihukm;h . h |u|k+1;h, instead of

k+1−m ku − Ihukm;h ≤ ch |u|k+1;h.

The statement of the equivalence of two norms becomes

k·kX ' k·kY , while

k·kX . k·kY states that the Y -norm is not weaker than the X-norm.

4.3.3 Definition: Let Th be a mesh on the domain Ω and Vh a finite element space with basis {ϕi}. Then, the matrix associated with the L2-inner product, Z mij = ϕj(x)ϕi(x) dx, Ω is called mass matrix. The matrix associated with the bilinear form of the equation is called stiffness matrix,

aij = a(ϕj, ϕi).

79 4.3.4 Lemma: Let {ϕi} be the basis of a ﬁnite element shape function space on a quasi-uniform family of meshes {Th}. Let Mh be the mass matrix. Then,

Λ(M) ' hd ' λ(M)

Therefore, the condition number is (asymptotically in h)

κ(M) ' 1.

Proof. For any mesh cell T ∈ Th, let xT be the entries of the vector x which belong to node values of the cell T . Let MT be the cell mass matrix obtained by restricting the L2-inner product to T . Then,

 T xT MT xT P 2 2 ≥ min |xT | ≥ λ(MT )|x| ,  |xT | T ∈Th 2 T X T T ∈Th kuhkL2(Ω)x Mhx = xT MT xT T xT MT xT P 2 2 ≤ max |xT | ≤ cΛ(MT )|x| , T ∈Th  |xT | T ∈Th T ∈Th where the constant in the upper bound is due to degrees of freedom shared by diﬀerent elements. Dividing by |x|2, we obtain

xT M x λ(M ) ≤ h ≤ cΛ(M ). (4.32) T |x|2 T

In order to estimate the eigenvalues of MT , we note that for a unisolvent element, the norms |xT | and kuk0,T are equivalent on the reference cell, and the L2-norm scales with hd when transforming to the real cell T . Thus, we have d λ(Mh) = O(h ) = Λ(Mh).

4.3.5 Corollary: Let Th be a shape-regular mesh with cell sizes ranging between the minimum h and the maximum H. Then, we have

Λ(M) ' Hd λ(M) ' hd H d κ(M) ' h

Remark 4.3.6. We have only computed the dependence of the condition number on the mesh size h, not on the polynomial degree of the shape function space. Indeed, the used equivalence between the L2-norm of a polynomial on the reference cell and the Euclidean norm of its coeﬃcient vector depends not only on the degree of the shape function space, but on its basis.

80 4.3.7 Theorem: Let {ϕi} be the nodal basis of a ﬁnite element shape function space on a quasi-uniform family of meshes {Th}. Let Ah be the stiﬀness matrix of the Laplacian with Dirichlet boundary values. Then,

Λ(A) ' hd−2, λ(A) ' hd

Therefore, the condition number is

κ(M) ' h−2.

Proof. First, we investigate the largest eigenvalue. By the scaling argument, d−2 there holds aij ' h . Since by the Gershgorin theorem all eigenvalues are bounded by the sum of the elements in a row, and this number is bounded d−2 uniformly on shape regular meshes, we have Λ(Ah) . h . Now, we choose a particular example, namely a vector x with a single one and all other entries zero. The corresponding ﬁnite element function uh has maximum one and support of radius h. Therefore, Z T 2 d−2 x Ahx = |∇uh| dx ' h . Ω

d−2 Thus, Λ(Ah) & h . In order to estimate the smallest eigenvalue, we use the Rayleigh quotient and the ﬁnite element function uh corresponding to the coeﬃcient vector x to obtain

T T 2 x Ahx x Ahx (uh, uh)L λ(Ah) = min = min n T n T x∈R x x x∈R (uh, uh)L2 x x a(u , u ) xT M x a(u , u ) xT M x = min h h h ≥ min h h min h . (4.33) n T 2 n T x∈ (u , u ) x x uh∈Vh x∈ x x R h h (uh, uh)L R

Thus, the smallest eigenvalue of Ah is bounded by the product of the smallest eigenvalue of the mass matrix and a value µ given by

a(u , u ) a(u, u) min h h ≥ min =: µ. (4.34) u ∈V 2 u∈V 2 h h (uh, uh)L (u, u)L Here, we need a result from the spectral analysis of the Laplacian. When we deﬁne the variational eigenvalue problem

1 a(u, v) = λ(u, v) ∀v ∈ V = H0 (Ω), (4.35) then, there is a positive minimal solution µ, the ﬁrst eigenvalue of the Lapla- cian, and an associated eigenfunction u, such that the equation holds for all v. −d Therefore, the eigenvalue is bounded with λ(Ah) & h . Furthermore, it can be

81 shown that this function is smooth and does not oscillate. This actually follows from the regularity result in Theorem 1.4.6. Therefore, the solution uh ∈ Vh to

a(uh, vh) = a(u, vh) ∀vh ∈ Vh, (4.36) converges to u in H1 and L2. With our standard estimates, we obtain

(u, uh) = (uh, uh) + (u − uh, uh) = (uh, uh) + rh(u), (4.37) where rh(u) → 0 as h → 0. Hence,

a(uh, uh) = a(u, uh) = µ(u, uh) → µ(uh, uh). (4.38) We conclude that the lower bound for the eigenvalue is sharp and the theorem holds.

4.4 Multigrid methods

Example 4.4.1. Smoothing property of the Richardson method with ω = 1/2 for the one-dimensional Laplacian.

k 4.4.2 Algorithm (The Multigrid iteration MGM(`, u` , b`)):

k,0 k 1. Pre-smoothing: let u` = u` and for i = 0, . . . , ν1 − 1 do

k,i+1 k,i k,i u` = u` + ω b` − A`u` . (4.39)

k,ν1 2. Coarse grid correction: compute b`−1 = r(b` − A`u` ) and µ (a) If ` = 1 solve A0u0 = b0 exactly 0 (b) If ` > 1, let u`−1 = 0 and for i = 0, . . . , µ − 1 do

i+1 i u`−1 = MGM(` − 1, u`−1, b`−1).

k,ν1+1 k,ν1 µ Then, add u` = u` + pu`−1.

3. Post-smoothing: for i = ν1 + 1, . . . , ν1 + ν2 do

k,i+1 k,i k,i u` = u` + ω b` − A`u` . (4.40)

k+1 k,ν1+ν2+1 and let u` = u` .

Remark 4.4.3. The multigrid method for µ = 1 is called the V-cycle, for µ = 2 it is the W-cycle. The generalization where µ depends on `, typically in the form 2L−`, where L is the ﬁnest level, is called the variable V-cycle.

82 Remark 4.4.4. The Richardson method in pre- and post-smoothing can be replaced by other relaxation methods like Jacobi, Gauss-Seidel, SOR, or SSOR. Even more complex operations may be considered.

If the smoother, like SOR, is not symmetric, it is possible to apply the transpose (backward SOR instead of forward) in post-smoothing to obtain a method which is symmetric.

4.4.5 Lemma: Let S` be the error propagation operator of the smoother. Then, the error propagation operator of a single step of the multigrid iteration is deﬁned recursively by

ν2 µ ν1 I − M` = S` (I − pM`−1rA)S` , (4.41)

−1 and M0 = A .

Proof. We prove for k = 0, which proves for every step and omit the superscript k. First, observe that for the exact solution u` of A`u` = b` there holds

i+1 i+1 u` − u` = S u` − u` i = i = 0, . . . , ν1 − 1, ν1 + 1, . . . , ν1 + ν2. Thus,

ν1 ν1 0 ν1+ν2+1 ν2 ν1+1 u` − u` = S (u` − u` ), u` − u` = S (u` − u` ), We continue by induction. For ` = 1, we have

ν1+1 ν1 µ ν1 −1 ν1 u1 − u1 = u1 − u1 − pu0 = u1 − u1 − pA0 rA1(u1 − u1 ). Combining with the error propagation of pre and post smoothing, we get

ν1+ν2+1 ν2 −1 ν1 0 u1 − u1 = S` (I − pA0 rA1)S` u1 − u1 .

83 Chapter 5

Discontinuous Galerkin methods

5.1 Nitsche’s method

5.1.1. Let us consider the inhomogeneous Dirichlet boundary value problem

−∆u = f in Ω (5.1) u = uD on ∂Ω.

We already know that for uD ≡ 0 we can discretize this problem by choosing a 1 D ﬁnite element space Vh ⊂ H0 (Ω). For general u , there are two options:

1. Compute an arbitrary “lifting” uD ∈ H1(Ω) such that it is equal to uD on D 1 the boundary and compute w = u − u ∈ H0 (Ω) as the solution to the weak formulation Z Z Z ∇w · ∇v dx = fv dx + ∇uD · ∇v dx. (5.2) Ω Ω Ω

D D 2. Compute an interpolation or projection uh of the boundary data u . Then, eliminate each row of the matrix corresponding to a degree of freedom on the boundary. In particular, let i be the index of such a degree of freedom and let k be an index corresponding to an interior degree of freedom not constraint by a boundary condition, but such that aik 6= 0 6= aki. Then, replace the rows

··· + a u + ··· + a u + ··· = f ii i ik k i (5.3) ··· + akiui + ··· + akkuk + ··· = fk

84 by the rows

D ui = ui D (5.4) ··· + 0 + ··· + akkuk + ··· = fk − akiui

The first option introduces the complication of finding a function in H1(Ω), which cannot be achieved automatically. The second can be implemented in an automatic way, but it complicates code, in particular for nonlinear problems. A completely different approach modifying the bilinear form was first suggested in the 60s and then perfected by Joachim Nitsche in 1971. In this section, we motivate this method and derive its error estimates. Its key feature is the 1 1 transition from Vh ⊂ H0 (Ω) to Vh ⊂ H (Ω). 5.1.2. If we simply derive our weak formulation in H1(Ω), we end up with an additional boundary term Z Z Z ∇u · ∇v dx = − ∆uv dx + ∂nuv ds. (5.5) Ω Ω ∂Ω

Thus, we obtain the natural boundary condition ∂nu = 0, which is not consistent with the original BVP. The first step for deriving Nitsche’s method is subtracting this boundary term on both sides. The result is the equation Z Z Z 1 ∇u · ∇v dx − ∂nuv ds = fv dx ∀v ∈ H (Ω). (5.6) Ω ∂Ω Ω We observe that the left hand side vanishes for any constant function u. Thus, we do not have unique solvability and we will have to fix this problem. Further- more, the boundary data uD does not appear in this formulation. We enforce u = uD in our formulation by a so called “penalty term” with penalty parameter α, modifying (5.6) to Z Z Z ∇u · ∇v dx − ∂nuv ds + αuv ds Ω ∂Ω ∂Ω Z Z = fv dx + αuDv ds ∀v ∈ H1(Ω). (5.7) Ω ∂Ω Integrating by parts, we see that u is a solution to this weak formulation. Follow- ing Nitsche, we make one additional modification which restores the symmetry of our form. We obtain the weak formulation

1 ah(u, v) = fh(v) ∀v ∈ H (Ω), (5.8) where Z Z Z Z ah(u, v) = ∇u · ∇v dx − ∂nuv ds − ∂nvu ds + αuv ds, Ω ∂Ω ∂Ω ∂Ω Z Z Z (5.9) D D fh(v) = fv dx − ∂nvu ds + αu v ds. Ω ∂Ω ∂Ω We abbreviate this equation to

85 Remark 5.1.3. Unfortunately, the problem (5.9) is not well-posed for any ﬁnite parameter α. Thus, it cannot be used to determine u ∈ H1(Ω). Nevertheless, we can establish well-posedness on discrete spaces Vh in order to compute a discrete solution uh and use the fact that u is already determined by the continuous problem. Our immediate goals are thus:

1. Establish the assumptions of the Lax-Milgram theorem on Vh, which in this case involves a suitable new norm for measuring the error. 2. Establish a relation between the discrete and continuous solution replacing Galerkin orthogonality. 3. Deriving error estimates in suitable norms. Notation 5.1.4. From now on, we will use the inner product notation Z Z (u, v) ≡ uv dx (∇u, ∇v) ≡ ∇u · ∇v dx, Ω Ω on Ω as well as Z hu, vi ≡ uv ds, ∂Ω on its boundary. Deﬁnition 5.1.5. A discrete problem (5.9) is called consistent, if for the solution u of the BVP (5.1) there holds

ah(u, vh) = fh(vh) ∀vh ∈ Vh. (5.10) Corollary 5.1.6. Let u ∈ H1(Ω) be the weak solution to (5.1) in the sense of (5.2). Then, the discrete problem (5.9) is consistent.

1 Proof. Since u ∈ H (Ω) and vh ∈ Vh, all boundary terms in ah(u, v − h) and D fh(vh) are well-defined and consistency follows from u = u in the sense of L2(∂Ω). Definition 5.1.7. The problem dependent norm used for the analysis of Nitsche’s method is defined by |||v|||2 = (∇v, ∇v) + hαv, vi. (5.11)

This choice is justiﬁed by

1 Lemma 5.1.8. Let Vh ⊂ V = H (Ω) be a piecewise polynomial ﬁnite element space on a shape-regular mesh Th. Then, if α suﬃciently large, there exist constants M and γ such that

ah(uh, vh) ≤ M|||uh||||||vh||| 2 ah(uh, uh) ≥ γ|||uh||| .

86 Proof. Key for the proof is the inverse trace estimate

−1/2 |v|H1(∂T ) ≤ ch |v|H1(T ), which holds with a constant c depending on shape regularity and polynomial degree. Thus, for a cell T at the boundary, there holds

−1/2 |h∂nuh, vhi∂T ∩∂Ω| ≤ ch |uh|H1(T )kvhkL2(∂T ∩∂Ω) 1 2 c2 ≤ |uh| 1 + kvhk 2 , 4 H (T ) hT L (∂T ∩∂Ω) We apply this to the lower bound to obtain

1 2 2c2 2 ah(uh, uh) ≥ 1 − |uh| 1 + α − kuhk 2 . 2 H (Ω) hT L (∂Ω)

Choosing

α 4c2 α(x) = 0 ≥ , (5.12) h(x) h(x) where h(x) is the size of the cell such that x ∈ ∂T , we obtain 1 a (u , u ) ≥ |||u |||2. h h h 2 h The proof of the upper bound follows the same fashion.

Corollary 5.1.9. Let α be chosen according to equation (5.12). Then, the discrete problem (5.8) has a unique solution uh ∈ Vh.

Proof. According to the previous lemma, the lemma of Lax-Milgram applies to the bilinear form ah(., .). For the right hand side fh(., ), we have again because 1 of trace estimates in H (Ω) and inverse estimates in Vh

D fh(v) ≤ c kfk0,Ω + ku kH1(Ω) |||v|||.

5.1.10 Theorem: Let u ∈ Hk+1(Ω) with k ≥ 1 be the solution of (5.1) and let uh ∈ Vh be the solution to (5.8) and let the assumptions of Lemma 5.1.8 hold. Let furthermore {Th} be a family of quasi-uniform, shape-regular meshes of maximal cell diameter h, and let the shape function spaces contain the polynomial space Pk. Then, there holds

k |||u − uh||| ≤ ch |u|k+1,Th . (5.13)

87 Proof. We begin with the triangle inequality

|||u − uh||| ≤ |||u − Ihu||| + |||Ihu − uh|||.

The interpolation error can be estimated by

k |u − Ihu|1,Ω ≤ h |u|2,Ω , 3/2 |u − Ihu|0,∂Ω ≤ h |u|2,Ω . (5.14) 1/2 |u − Ihu|1,∂Ω ≤ h |u|2,Ω . The second and third estimate actually require some deeper arguments from functional analysis, which is beyond the scope of this class. It involves an intuitive notion of Sobolev spaces with non-integer derivatives. Allowing such spaces, the trace estimate becomes

kuk1/2,∂Ω ≤ ckuk1,Ω. (5.15)

For the remaining error term, we use Vh-ellipticity of the discrete form and consistency to obtain

2 γ|||Ihu − uh||| ≤ ah(Ihu − uh,Ihu − uh) = ah(Ihu − u, Ihu − uh). (5.16)

Using Young’s inequality, we estimate the right hand side with εh = u − Ihu and ηh = Ihu − uh on each boundary cell T with boundary edge E by

1 2 γ 2 |(∇εh, ∇ηh)T | ≤ γ |εh|1,T + 4 |ηh|1,T , 2 √ 2 γ √ 2 |hαεh, ηhiE| ≤ γ k αεhk0,E + 8 k αηhk0,E √ q (5.17) 1 2 γ 2 γ α0 2 |hεh, ∂nηhi | ≤ k αεhk + |ηh| + k ηhk E γ 0,E 4 1,T 4 hT 0,E

2hT 2 γα0 2 |h∂nεh, ηhi | ≤ |εh| + kηhk . E γα0 0,E 8hT 0,E Adding these over all cells, we obtain

2 γ|||Ihu − uh||| √ γ 2 1 2 3 2 2hT 2 ≤ |||Ihu − uh||| |u − Ihu| + k α(u − Ihu)k + |u − Ihu| , 2 γ 1,Ω γ 0,E γα0 0,E and thus, by the interpolation estimate (5.14)

2 2 2k 2 |||Ihu − uh||| ≤ γ ch |u|k+1,Ω.

88 5.1.11 Theorem: Assume in addition to the assumptions of Theo- rem 5.1.10 that the adjoint problem

−∆z = u − u in Ω h (5.18) z = 0 on ∂Ω.

admits elliptic regularity, namely

|z|2 ≤ cku − uhk0. (5.19)

Then, the solutions u and uh admit the estimate

k+1 ku − uhk0 ≤ ch |u|k+1. (5.20)

Proof. Due to symmetry of the discrete bilinear form, we have adjoint consistency:

ah(vh, z) = (u − uh, vh) ∀vh ∈ Vh. (5.21)

From here, we proceed like in the continuous case:

2 ku − uhk0 = ah(u − uh, z) = ah(u − uh, z − Ihz).

Using the same derivation as in the previous theorem, we obtain the result. Remark 5.1.12. We chose the norm deﬁned in (5.11) for our energy norm analysis in Theorem 5.1.10. We could have done the same using the operator p norm ah(v, v). In fact, Lemma 5.1.8 states that both norms are equivalent. Therefore, we chose the one involving less terms and providing for simpler interpolation estimates.

The “triple norm” notation |||.||| is very common as a notation for problem ad- justed norms.

5.2 The interior penalty method

5.2.1. We review the basic deﬁnitions necessary to describe discontinuous Galerkin (DG) methods. In particular, we need the sets of faces Fh of a mesh, discontinuous piecewise polynomial spaces and broken integrals.

89 d 5.2.2 Deﬁnition: Let Th be a mesh of Ω ⊂ R consisting of mesh cells a Ti. For every boundary facet F ⊂ ∂Ti, we assume that either F ⊂ ∂Ω or F is a boundary facet of another cell Tj. In the second case, we indicate this relation by labeling this facet Fij. The set of all facets Fij is the set i ∂ of interior faces Fh. The set of facets on the boundary is Fh. aThis assumption can indeed be relaxed

5.2.3 Deﬁnition: The discontinuous ﬁnite element space on Th is constructed by concatenation of all shape function spaces PT for T ∈ Th without additional continuity requirements:

2 Vh = v ∈ L (Ω) v|T ∈ PT ∀T ∈ Th . (5.22)

5.2.4 Deﬁnition: For any set of cells Th or faces Fh, we deﬁne the bilinear forms X (u, v) = (u, v) , (5.23) Th T T ∈Th X hu, vi = hu, vi . (5.24) Fh F F ∈Fh (5.25)

5.2.5. We start out with the equation

−∆u = f.

Integrating by parts on each mesh cell yields

(−∆u, v)T = (∇u, ∇v)T − h∂nu, vi∂T = (f, v)T . We realize that the choice of discontinuous ﬁnite element spaces introduces a consistency term on the interfaces between cells and on the boundary.

On interior faces, there is the issue that u and ∂nu actually have two values on the interface, one from the left cell and one from the right. Therefore, we have to consolidate these two values into one. To this end, we introduce the concept of a numerical ﬂux, which constructs a single value out of these two. Thus, we introduce on the interface F between two cells T + and T − ∇u+ + ∇u− F(∇u) = =: {{∇u}}. 2

90 Using h∂nu, vi = h∇u, vni we change our point of view and instead of integrating over the boundary ∂T , we integrate over a face F between two cells T + and T −. Adding up integrals from both sides, we obtain the term

+ + − − − {{∇u}}, v n + v n F = −2h{{∇u}}, {{vn}}iF . On boundary faces, we simply get

h∂nu, viF .

Adding over all cells and faces, we obtain the equation

(∇u, ∇v) − 2h{{∇u}}, {{vn}}i i − h∂nu, vi ∂ = (f, v) . Th Fh Fh Ω

Following the idea of Nitsche, we symmetrize this term to obtain

(∇u, ∇v) − 2h{{∇u}}, {{vn}}i i − 2h{{un}}, {{∇v}}i i Th Fh Fh o − h∂nu, vi ∂ − hu, ∂nvi ∂ = (f, v) − hu , ∂nvi ∂ . Fh Fh Ω Fh Here the second term on the right was introduced for consistency. Finally, it turns out that this method is not stable and needs stabilization by a jump term. This will be done in Deﬁnition 5.2.8. Before, we introduce the notation for averaging and jump operators.

5.2.6 Notation: Let F be a face between the cells T + and T −. Let n+ and n− = −n+ be the outer normal vectors of the cells at a point x ∈ F . + For a function u ∈ Vh, the traces u and u− of u on F taken from the cell T + and T − are deﬁned as:

u+(x) = lim u(x − εn+), ε&0 u−(x) = lim u(x − εn−). ε&0

We deﬁne the averaging operator {{.}} and the jump operator . as J K u+ + u− {{u}} = , u = u+ − u−. (5.26) 2 J K Not that the sign of the jump of u depends on the choice of the cells T + and T −. It will only be used in quadratic terms.

Remark 5.2.7. The jump can be denoted as the mean value of the product of a function and the normal vector,

u = 2{{un}}· n+ = −2{{un}}· n−. (5.27) J K

91 5.2.8 Deﬁnition: The interior penalty methoda uses the bilinear form

ah(u, v) = (∇u, ∇v) + hσh u , v i i + hσhu, vi ∂ Th J K J K Fh Fh − 2h{{∇u}}, {{vn}}i i − 2h{{un}}, {{∇v}}i i Fh Fh

− h∂nu, vi ∂ − hu, ∂nvi ∂ , (5.28) Fh Fh and the linear form

D D fh(v) = (f, v)Ω − u , ∂nv ∂ + σhu , v ∂ , (5.29) Fh Fh where f is the right hand side of the equation and uD the Dirichlet boundary value.

aAlso known as symmetric interior penalty (SIPG) or IP-DG.

5.2.9 Deﬁnition: On the space Vh we deﬁne the norm k.k1,h by

2 X 2 X √ 2 X √ 2 kvk1,h = k∇vkT + k σh v kF + k σhvkF . (5.30) T ∈ i J K ∂ Th F ∈Fh F ∈Fh

5.2.10 Problem: Prove that the norm deﬁned in (5.30) is indeed a norm on Vh.

5.2.11 Lemma: Let Th be shape-regular and chosen on each face F as σh = σ0/hF , where hT is the minimal diameter of a cell adjacent to F . Then, there is a σ0 > 0 such that there exists a constant γ > 0, such that independent of h there holds

2 ah(uh, uh) ≥ γkuhk1,h ∀uh ∈ Vh. (5.31)

5.2.12 Problem: Prove Lemma 5.2.11.

92 5.2.13 Lemma: Let f ∈ L2(Ω) and let the boundary conditions admit that for the solution to

−∆u = f in Ω, u = uD on ∂Ω,

there holds u ∈ H1+ε(Ω) for a positive ε. Then, the interior penalty method is consistent, that is,

ah(u, vh) = fh(vh) ∀vh ∈ Vh. (5.32)

Proof. From f ∈ L2(Ω) we deduce that ∇u ∈ Hdiv(Ω). Thus, with the extra regularity, the traces of ∂nu on faces are well-deﬁned and coincide from both sides. The remainder is integration by parts.

s+1 5.2.14 Theorem: For k ≥ 1 let Pk ⊂ PT and u ∈ H (Ω) with 1/2 ≤ s ≤ k. Then, the interior penalty method admits the error estimate

s ku − uhk1,h ≤ ch |u|s+1. (5.33)

If furthermore the boundary condition admits elliptic regularity, there holds

s+1 ku − uhk0 ≤ ch |u|s+1. (5.34)

5.2.1 Bounded formulation in H1

5.2.15. The interior penalty method introduced so far is Vh-elliptic and consistent, but it is not bounded on H1(Ω). This was a reason, why we could not use standard techniques for the proof of the convergence result and after applying consistency had to estimate each term separately.

In this section, we will introduce a reformulation of the interior penalty method, 1 which is equivalent to the original method on Vh, but is also bounded in H (Ω). As an unpleasant side eﬀect, it turns out that this method is inconsistent, and we have to estimate the consistency error.

The main technique applied here is the use of lifting operators, such that the traces of derivatives on faces can be replaced by volume terms. Note that the lifting operators, while very useful for the analysis of the method, are not actually used in the implementation of the interior penalty method.

93 5.2.16 Deﬁnition: Deﬁne the auxiliary space

2 d Σh = τ ∈ L (Ω; R ) ∀T ∈ Th : τ|T ∈ ΣT , (5.35)

where ΣT is a (possibly mapped) polynomial space chosen such that ∇VT ⊂ ΣT . Then, we deﬁne the lifting operator

L : V + Vh → Σh (5.36)

(L v, τ) = 2h{{τ}}, {{vn}}i i + hτ · n, vi ∂ . (5.37) Th Fh Fh

2 5.2.17 Lemma: The lifting operator is a bounded operator from L (Fh) to Σh, such that

1 1 kL vkL2(Ω) ≤ c √ v + √ v . (5.38) h i h ∂ J K Fh Fh In particular, it is bounded on H1(Ω).

2 Proof. It is clear, that the operator is bounded on L (Fh), since its deﬁnition involves face integrals weighted with polynomial functions. The dependence on the mesh size is due to the standard scaling argument.

5.2.18 Deﬁnition: The interior penalty method with lifting operators uses the bilinear form

ah(u, v) = (∇u, ∇v) − (L u, ∇v) − (∇u, L v) Th Th Th

+ hσh u , v i − + hσhu, vi ∂ . (5.39) J K J K Fh Fh and the linear form (5.29) of the original interior penalty method. Its residual operator is

Res(u, v) = ah(u, v) − (f, v). (5.40)

5.2.19 Lemma: The interior penalty method in flux form (Defini- tion 5.2.8) and in lifting form (Definition 5.2.8) coincide on the discrete space Vh if Σh is chosen such that ∇Vh ⊂ Σh.

94 Proof. Since ∇Vh ⊂ Σh, ∇uh and ∇vh are valid test functions in the deﬁni- tion (5.37) of the lifting operator, and the equality

(L uh, ∇vh) = 2h{{uhn}}, {{∇vh}}i i + huh, ∂nvhi ∂ . Th Fh Fh

5.2.20 Deﬁnition: Let V ⊂ H1(Ω) and let u, u∗ ∈ V solve the primal and dual problems

a(u, v) = f(v), a(v, u∗) = ψ(v), ∀v ∈ V, (5.41)

with a bounded, V -elliptic bilinear form a(., .). For a discrete bilinear form ah(., .) deﬁned on V + Vh, we deﬁne the primal and dual residual operators

Res(u, v) = ah(u, v) − f(v), ∗ ∗ ∗ (5.42) Res (u , v) = ah(v, u ) − ψ(v).

5.2.21 Lemma: Let ah(., .) be a bounded bilinear form on V + Vh and

elliptic on Vh with norm k.kVh and constant γ. Then, the error u − uh admits the estimate 1 kahk ku − u k ≤ kRes(u, .)k ∗ + 1 + inf ku − w k (5.43) h Vh Vh h γ γ wh∈Vh

Proof. First, by the deﬁnition of the residual, we have the error equation

ah(u − uh, vh) = Res(u, vh), ∀vh ∈ Vh. (5.44)

Inserting wh − wh for an arbitrary element wh ∈ Vh, we obtain

ah(wh − uh, vh) = Res(u, vh) − ah(u − wh, vh), ∀vh ∈ Vh.

Using vh = wh − uh and ellipticity, we obtain γkw − u k2 ≤ a (w − u , w − u ) h h Vh h h h h h = Res(u, wh − uh) − ah(u − wh, wh − uh) ≤ kRes(u, .)k ∗ + ka kku − w k kw − u k . Vh h h Vh h h Vh Hence, by triangle inequality 1 kahk ku − u k ≤ kRes(u, .)k ∗ + 1 + inf ku − w k h Vh Vh h Vh γ γ wh∈Vh

95 5.2.22 Lemma: Let u ∈ V be the solution to the Poisson equation with right hand side f ∈ L2(Ω). Assume u ∈ Hs(Ω) with s > 3/2. Then, we have for v ∈ V + Vh:

(f, v) = (∇u, ∇v) − 2h∇u, {{vn}}i i − h∂nu, vi ∂ . (5.45) Th Fh Fh

Proof. We set out from the strong form of the Poisson equation and integrate by parts. X (f, v) = (−∆u, v) = (∇u, ∇v) − h∂nu, vi . Th ∂T T ∈Th Under the regularity assumptions of the lemma, all of these integrals make sense 2 at least as duality pairings. In particular, ∂nu ∈ L (∂T ), and thus we can split ∂T into individual faces. Therefore, X h∂nu, vi = 2h∇u, {{v ⊗ n}}i i + h∂nu, vi ∂ . ∂T Fh Fh T ∈Th The proof concludes by collecting the results.

5.2.23 Lemma: Let k ≥ 1 and let Vh such that Pk−1 ⊂ ΣT . Then, if k+1 u ∈ H (Ω) and v ∈ V + Vh, there holds

k √ √ |Res(u, v)| ≤ ch |u|k+1 k σh v k i + k σhvk ∂ Fh Fh (5.46) k J K ≤ ch |u|k+1kvk1,h.

Proof. First, we observe that by the regularity assumption, u = 0 and thus, L u = 0. Hence, J K

ah(u, v) = (∇u, ∇v) − (∇u, L v) . Th Th By Lemma 5.2.22 and regularity of u,

Res(u, v) = 2h∇u, {{vn}}i i + h∂nu, vi ∂ − (∇u, L v) Fh Fh Th

= 2h{{∇u}}, {{vn}}i i + h∂nu, vi ∂ − (∇u, L v) Fh Fh Th

= 2h{{∇u}}, {{vn}}i i + h∂nu, vi ∂ − (ΠΣh ∇u, L v) , Fh Fh Th 2 where ΠΣh is the L -projection. Now, we can apply the deﬁnition of the lifting term to obtain

D 1 E Res(u, v) = 2 {{∇u − ΠΣ ∇u}}, σh{{vn}} σh h i Fh D 1 E + (∇u − ΠΣ ∇u) · n, σhv . σh h ∂ Fh

96 Application of standard approximation and trace estimates yields the result observing that σh = σ0/h.

5.2.24 Theorem: Let k ≥ 1 and Vh such that Pk ⊂ VT . Let u ∈ k+1 H (Ω) be the solution to the continuous Poisson problem. Let ah(., .) be the interior penalty method with lifting operators such that ∇Vh ⊂ Σh. Then, there holds

k ku − uhk1,h ≤ ch |u|k+1. (5.47)

Proof. Application of Lemma 5.2.21, Lemma 5.2.23, and standard interpolation results.

5.2.25 Theorem: Let the assumptions of Theorem 5.2.24 hold and in addition assume that the problem

a(v, u∗) = ψ(v), ∀v ∈ V,

admits the elliptic regularity estimate

∗ ku kH2(Ω) ≤ ckψkL2(Ω). (5.48)

Then, there holds

k+1 ku − uhkL2(Ω) ≤ ch |u|Hk+1(Ω). (5.49)

Proof. The proof uses the duality argument by Aubin and Nitsche, which sets out solving the auxiliary problem

∗ a(v, u ) = (u − uh, v), ∀v ∈ V.

Using the deﬁnition of the dual residual, we obtain the equation

∗ ∗ ∗ (u − uh, v) = ah(v, u ) − Res (u , v), ∀v ∈ V + Vh.

Testing with v = u − uh yields

2 ∗ ∗ ∗ ∗ ku − uhk = ah(u − uh, u ) − Res (u , u − u ).

Additionally, we us the error equation

ah(u − uh, vh) = Res(u, vh),

97 ∗ tested with vh = Ihu , to obtain

2 ∗ ∗ ∗ ∗ ∗ ku − uhk = ah(u − uh, u − Ihu ) − Res (u , u − uh) + Res(u, Ihu ).

Using the regularity of u∗, the ﬁrst term on the right admits the estimate

∗ ∗ ∗ ∗ |ah(u − uh, u − Ihu )| ≤ ku − uhk1,hku − Ihu k1,h ≤ chku − uhk1,h.

For the second term, we use Lemma 5.2.23 to obtain

∗ ∗ ∗ |Res (u , u − uh)| ≤ ch|u |2ku − uhk1,h.

Finally, using u∗ = 0, the same lemma yields J K ∗ √ ∗ √ ∗ |Res(u, Ihu )| ≤ ch|u|2 k σh Ihu k i + k σhIhu k ∂ Fh Fh √ J ∗ K ∗ √ ∗ ∗ = ch|u|2 k σh u − Ihu k i + k σh(u − Ihu )k ∂ Fh Fh k ∗ J K ≤ ch|u|2h |u |k+1

Using the energy estimate in Theorem 5.2.24 we can conclude the prove.

98 Bibliography

[Ciarlet, 1978] Ciarlet, P. G. (1978). The Finite Element Method for Elliptic Problems. North-Holland.

[Evans, 1998] Evans, L. C. (1998). Partial Diﬀerential Equations. American Mathematical Society. [Gilbarg and Trudinger, 1998] Gilbarg, D. and Trudinger, N. S. (1998). Elliptic Partial Diﬀerential Equations of Second Order. Springer.

[Kondrat’ev, 1967] Kondrat’ev, V. A. (1967). Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc., 16:227–313. [Rannacher and Scott, 1982] Rannacher, R. and Scott, L. R. (1982). Some optimal error estimates for piecewise linear ﬁnite element approximations. Math. Comput., 38:437–445.

[Schatz and Wahlbin, 1977] Schatz, A. H. and Wahlbin, L. B. (1977). Maximum norm estimates in the ﬁnite element method in plane polygonal domains. Part I. Math. Comput., 32:73–109. [Verfürth, 2013] Verfürth, R. (2013). A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press.

99 Index

V -elliptic, 17 Continuous basis functions, 33 [Gilbarg and Trudinger, 1998, Theorem continuously embedded, 22 8.8], 24 data oscillation, 55 a posteriori error estimator, 50 degrees of freedom, 28 adjoint problem, 47 Dirac δ-distribution, 21 affine mapping, 37 Dirichlet energy, 6 approximation error, 60 Dirichlet principle, 5 Assembling the matrix, 36 Dirichlet problem, 5, 6, 10 averaging operator, 91 Discontinuous basis functions, 34 discrete problem, 35 Banach fixed-point theorem, 71 distributional derivative, 20 Banach space, 13 domain, 5, 69 barycentric coordinates, 30, 52, 56 dual problem, 47 bilinear form, 17 dual space, 16 bilinear mapping, 37 boundary, 5 edge, 27 Bramble-Hilbert, 41 efficient, 50 broken Sobolev norm, 43 elliptic, 17, 19, 69 bubble function, 56 Elliptic regularity, 47 Bunyakovsky-Cauchy-Schwarz inequal- elliptic regularity, 47, 48, 93, 97 ity, 11, 13, 16 equivalent, 45 error propagation operator, 71 Céa, 36 essential boundary condition, 10 Cauchy sequence, 9, 13 cell, 27 face, 27 Clément quasi-interpolation, 51 facet, 27 closed, 13 finite element, 28 coercive, 17 finite element space, 28 complete, 13, 13 Friedrichs inequality, 8, 9 completion, 13 condition number, see spectral condi- Galerkin approximation, 35, 35 tion number Galerkin equations, 35 conforming approximation, 35 Global regularity, 25 conjugate gradient method, 75, 77 gradient, 4 consistency error, 60 Green’s function, 48 consistent, 86 Heaviside function, 20, 21

100 Hilbert space, 13 orthogonal projection, 15 homogeneous, 5 orthogonal set, 13 orthonormal set, 13 inner product, 11, 11 inner product space, 13 Poincaré inequality, 41 interior penalty method, 92, 94 Poisson’s equation, 4, 5 Interior regularity, 25 pre-Hilbert space, 13 Inverse estimate, 45 preconditioned cg method, 77 preconditioned Richardson iteration, 74 jump operator, 53, 91 preconditioner, 74 pull-back, 37 κ(A), 69 κ(A), 70 quasi-uniform, 43 Kondratev, 25 Krylov space, 77 Rayleigh quotient, 70 reference cell, 37 Λ(A), 69 relaxation parameter, 72 λ(A), 69 reliable, 50 Λ(B,A), 75 residual, 53 λ(B,A), 75 residual operator, 95 Λ(A), 70 Richardson’s method, 72 λ(A), 70 Riesz isomorphism, 16, 73, 75, 77 Laplacian, 4 Riesz representation theorem, 16, 72 Lax-Milgram, 18 lifting operator, 94 Scaling lemma, 39 linear functional, 16 Schöberl quasi-interpolation, 52 linear mapping, 16 Scott-Zhang quasi-interpolation, 52 locally quasi-uniform, 50 shape function, 28 shape function basis, 28 mass matrix, 79 shape regular, 43 mean value operator, 53 Sobolev space, 9, 21, 22 mesh, 27 spectral condition number, 69, 70, 71, Method of steepest descent, 75 78 Meyers-Serrin, 22 spectral equivalence, 74, 78 minimizing sequence, 9 steepest descent, 76 multigrid iteration, 82 stiﬀness matrix, 79 Strang’s ﬁrst lemma, 59 natural boundary condition, 10 stronger norm, 45 nodal interpolation, 43 symmetric, 69 node functional, 28 normal derivative, 5 tensor product basis, 31 normal vector, 5 tensor product polynomials, 31 k normed dual, 16 The Multigrid iteration MGM(`, u` , b`), normed vector space, 13 82 The space Q2, 33 orthogonal, 13 topology, 28 orthogonal complement, 13 Trace theorem, 23

101 unisolvent, 28 vertex, 27 weak derivative, 20 weaker norm, 45

102