Linear Finite Element Methods

Chapter 3

The ﬁnite element methods provide

spaces V of functions that are piecewise smooth and “simple”, and • n locally supported basis function of these spaces • to achieve

good approximations of the solution u V , • ∈ an eﬃcient assembly of the system matrix with desirable properties (e. g. sparse and/or well condi- • tioned).

3.1 Meshes

A mesh of the computational domain Ω Rd, d = 1, 2, 3, is a collection of non-overlapping • cells K MM , M := , that are open, simply⊂ connected subsets K Ω. { i}i=1 |M| i ⊂ Each cell K is a C∞-diﬀeomorphic image of a closed d-dimensional polytope K — the reference • i cell — which is a convex hull of at least d + 1 points in Rd. b Each cell K is an interval (d = 1), a Lipschitz polygon (d = 2), or a Lipschitz polyhedron (d = 3). • i The mesh contains a set of nodes (vertices) ( ), edges ( )(d > 1), and faces ( ) • (d = 3). (SometimesM the components of the meshN of dimensionM d E M1 are called faces). F M − The topology of the mesh is described by the connectivity of cells, faces, edges and nodes, e. g., • does a face belong to a cell, or more precisely, is a face contained in the closure of a cell.

A mesh is conforming if the intersection K K of (the closure of) any two adjacent cells K , • i ∩ j i Kj is a face of both cells. A node of a mesh that is located in the interior of a geometric face of one of its cells is known as hanging (dangling)M node.

Grids are meshes with translation invariant structure. These can be tensor product grids, • that are meshes whose cells are quadrilaterals (d = 2) or hexahedra (d = 3) with parallel sides.

Meshes

are a crucial building block in the design of the ﬁnite dimensional trial and test spaces used in the • ﬁnite element method, and

provide subdomains for integration to build the system matrix and vector of the right hand side. • The orientation of geometric objects of the mesh is important for deﬁning high order basis functions.

For an edge we have to deﬁne its direction. • For a face (d = 3) we have to specify an ordering of the edges along its boundary. •

25 (a) (b) (c)

Figure 3.1: (a) Two-dimensional mesh and the sets of edges (red) and nodes (blue). (b) A geometrical node is not a topological node. (c) A triangulation that is not conforming and possesses two hanging nodes.

Figure 3.2: Examples of triangular and quadrilateral grids in two dimensions.

Reference cell K

for triangles:b reference triangle with nodes (0, 0), (1, 0), (1, 1) • for quadrilaterals: reference quad [0, 1]2 or [ 1, 1]2 • − for tetrahedra: reference tetrahedron with nodes (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 1, 0) • for prism: reference prism = reference triangle [0, 1] • × for pyramid: reference pyramid with nodes (0, 0, 0), (1, 0, 0), (1, 1, 0), (0, 1, 0) and (0.5, 0.5, 1) • for hexahedra: reference hexahedron [0, 1]3 • In meshes with diﬀerent cell types, we have more than one reference cell.

(a) Triangle (b) Quadrilateral (c) Tetrahedron (d) Prism (e) Pyramid (f) Hexahedron

Figure 3.3: Cell types in 2D and 3D.

Aﬃne mapping: linear, bijective mapping from a reference cell (triangle/parallelogram) to a physical cell (triangle/parallelogram) in the mesh

d d d,d d Φ : R R , ξ Fξ + τ , F R regular , τ R . (3.1) → 7→ ∈ ∈ A mesh is called aﬃne equivalent if all its cells arise as aﬃne images of a single d-dimensional (reference) cell.

26 Automatic mesh generation: algorithms that create a mesh starting from a description of Ω which can be given

in terms of geometric primitives (ball, brick, etc.) whose unions or intersections constitute Ω, • by means of a parametrization of the faces of Ω, • through a function f : Rd R, whose sign indicates whether a point is located inside Ω or outside, • → by a mesh covering the surface of Ω and a direction of the exterior unit normal. • Various strategies can be employed for automatic grid generation:

advancing front methods, that build cells starting from the boundary, • Delaunay reﬁnement techniques, that can create a mesh starting from a mesh for ∂Ω or a “cloud” • of points covering Ω,

the quadtree (d = 2) or octree (d = 3) approach, which ﬁlls Ω with squares/cubes of diﬀerent sizes • supplemented by special measures for resolving the boundary, and

mapping techniques, that split Ω into subdomains of “simple” shape (curved triangles, parallelo- • grams, bricks), endow those with parametric grids and glue these together.

Remark 3.1. Traditional codes for the solution of boundary value problems based on the finite element method usually read the geometry from a file describing the topology and geometry of the underlying mesh. Then an approximate solution is computed and written to a file in order to be read by post-processing tools like visualization software, see Figure 3.4. Parameters

Finite element solver Post-processor Mesh generator (computational kernel) (e. g. visualization)

Figure 3.4: Flow of data in traditional ﬁnite element simulations

Remark 3.2. A typical ﬁle format for a mesh of a simplicial conforming triangulation of a two- dimensional polygonal domain is the following:

# Two-dimensional simplicial mesh N N # Number of nodes ∈ ξ1 η1 # Coordinates of first node

ξ2 η2 # Coordinates of second node . .

ξN ηN # Coordinates of N-th node M N # Number of triangles ∈ 1 1 1 n1 n2 n3 X1 # Indices of nodes of first triangle 2 2 2 n1 n2 n3 X2 # Indices of nodes of second triangle . . M M M n1 n2 n3 XM # Indices of nodes of M-th triangle

Here, Xi, i = 1,...,M, is an additional piece of information that may, for instance, describe what kind of material properties prevail in triangle #i. In this case Xi may be an integer index into a look-up table of material properties or the actual value of a coeﬃcient function inside the triangle.

27 Additional information about edges located on ∂Ω may be provided in the following form:

K N # Number of edges on ∂Ω ∈ 1 1 n1 n2 Y1 # Indices of endpoints of first edge 2 2 n1 n2 Y2 # Indices of endpoints of second edge . . K K n1 n2 YK # Indices of endpoints of K-th edge where Yk, k = 1,...,K, provides extra information about the type of boundary condition to be imposed on edge k. Note that the ordering of the nodes in the above ﬁle formats implies an orientation of triangles and edges. For a comprehensive account on mesh generation see [FG00]. An interesting algorithm for Delaunay meshing is described in [Rup95, She96]. Free mesh generation software is also available, just to name some, netgen, gmsh, triangle, emc2. However, the most sophisticated mesh generation tools are commercial products and their algorithmic details are classiﬁed.

3.2 Linear ﬁnite elements on triangular meshes 3.2.1 Basis functions

Figure 3.5: Finite element basis function bP (x).

Let the domain Ω R2 be a polygon with a triangular conforming mesh . We define the finite element space of piecewise∈ linear, continuous functions as M S1(Ω, ) := u C0(Ω) : u(x) = a + bx + cx , K . M { ∈ |K 1 2 ∀ ∈ M} Proposition 3.3 (Properties of S1(Ω, )). M It holds S1(Ω, ) H1(Ω). • M ⊂ u S1(Ω, ) is uniquely defined by the values of u(P ) on the nodes P ( ). • ∈ M ∈ N M N = dim S1(Ω, ) = < . • M |N | ∞ S1(Ω, ) = span b (x): P ( ) , with so called hat functions defined as • M { P ∈ N M } 1 0 b S (Ω, ), b (P ) = δ 0 . P ∈ M P P =P Let b be the vector of the basis functions. Then, an arbitrary FE function v S1(Ω, ) can be written as ∈ M > v(x) = v(P )bP (x) = v b(x), P ∈NX(M) so also the solution > un(x) = un(P )bP (x) = u b(x). P ∈NX(M)

28 3.2.2 Assembling of system matrix and load vector

The support of the basis functions bP (x) consists of only a few triangles, such that the integrals in the (bi-)linear forms reduce to smaller sets and vanish for many pairs of basis functions. Changing the point of view, we consider the shape functions, which are the restrictions of basis functions to one cell K . For S1(Ω, ) these are exactly three, each for one node of K. The shape functions∈ can M be deﬁned onM a single cell K as

−1 NK,Pj (K)(x) = Nj(ΦK x), where Nj are the element shape functions deﬁned byb

N (ξ) = 1 ξ ξ , N (ξ) = ξ , j = 1, 2, b 0 − 1 − 2 j j on the reference elementb K, which is the triangle with vertices (0b, 0), (1, 0), (0, 1). Pj(K) is the j-th node of the triangle K, j = 0, 1, 2, with the coordinate pj. Then, the (aﬃne) element mapping reads b x = Φ (ξ) = p + ξ (p p ) + ξ (p p ) = τ + F ξ, K 0 1 1 − 0 2 2 − 0 K with FK = (p1 p0, p2 p0) and τ = p0. As the element− mapping− is linear, the shape functions are in fact linear as well. The basis functions result from the shape functions by glueing, i. e.,

NK,Pj (x) x K,P (K),P = Pj(K), bP (x) = ∈ ∈ N (0 otherwise. This can be expressed by the connectivity matrices or T-matrices

3,N 1,Pj(K) = Pi( ), TK R , (TK )ij = M ∈ (0, otherwise. These T-matrices give a relation between the local number of (element) shape functions and global numbering of basis functions. They have the form ... 1 ...... · · · ··· ··· · TK = ...... 1 ......  · ... ··· ... · · ... ···1 ... ·  · ··· ··· · · ·   For the T-matrices an extremely sparse format is used. For linear ﬁnite elements on triangles we have only to store the three indices Pi0 , Pi1 and Pi2 of the three nodes in the triangle. Now, we can express a basis function in one cell K as

3 b (x) = (T) N (x), Pi |K ki K,Pk(K) kX=1 and the system matrix is given with

(B) = b(b , b ) = b (b , b ) ij Pj Pi K Pj |K Pi |K XK 3 3

= bK ( (TK )kjNK,Pk(K), (TK )`iNK,P`(K)) XK kX=1 X`=1 3 3

= (TK )kj(TK )`ibK (NK,Pk(K),NK,P`(K)) XK kX=1 X`=1 > = (TK )·,jBK (TK )·,i, XK i. e., as sum of “weighted” element matrices

> B = TK BK TK . XK This means we have to integrate only over the shape functions, and to sum up the contributions over all the cells.

29 Element stiﬀness matrix The bilinear form in the variational problem related to (1.4) consists of two parts, which contribute to the system matrix. The ﬁrst part

a(u, v) = a u, v dx h ∇ ∇ i ZΩ is constituting the stiﬀness matrix A. We transform the derivatives to K. By chain rule of diﬀerentiation and with

∂ ∂ > ∂ ∂ > = b , , = , , ∇ ∂x ∂x ∇ ∂ξ ∂ξ 1 2 1 2 we have b

∂x1 ∂x2 ∂ξ1 NK,Pj (x(ξ)) = ∂x1 NK,Pj (x) + ∂x2 NK,Pj (x) ∂ξ1 ∂ξ1 ∂x1 ∂x2 ∂ξ2 NK,Pj (x(ξ)) = ∂x1 NK,Pj (x) + ∂x2 NK,Pj (x) ∂ξ2 ∂ξ2 and thus ∂Φ > N (x(ξ)) = K N (x). ∇ K,Pj ∂ξ ∇ K,Pj > > b DΦK =FK Hence, for triangular cells we obtain | {z }

N (x) = F−> N (x(ξ)) = F−> N (ξ). ∇ K,Pj K ∇ K,Pj K ∇ j For a single cell K the entries of the elementb stiﬀness matrix AKbareb given by

a (N ,N ) = a(x)( N (x))> N (x)dx K K,Pj K,Pi ∇ K,Pj ∇ K,Pi ZK = a(Φ ξ)( N (ξ))>F−1F−> N (ξ) F dξ K ∇ j K K ∇ i | K | ZKb = a(Φ ξ)( b Nb (ξ))>adj(F )adj(b bF )> N (ξ) F −1 dξ, K ∇ j K K ∇ i | K | ZKb where we used the relation of the inverse andb theb adjoint matrix b b

F−1 = F −1adj(F ). K | K | K a b d b The adjoint matrix of a matrix is . c d c− a Note, that the gradient of the element shape− functions are constant vectors 1 1 0 N = − , N = , N = . ∇ 0 1 ∇ 1 0 ∇ 2 1 − So we can simplifyb b b b b b a (A ) = a (N ,N ) = K ( N )>adj(F )adj(F )> N F −1 K ij K K,Pj K,Pi 2 ∇ j K K ∇ i| K | with the average heat conduction a = 1 a(xb)dbx in K. (Note that bFb = 2 K ). The latter K |K| K | K | | | simpliﬁes for constant material or may be obtained by numerical quadrature for more general smooth R functions. Due to the special values of N we can even simplify ∇ j aK > b b AK = D DK 4 K K | | with a matrix DK with coordinate diﬀerences y y y y y y D = 1 2 2 0 0 1 . K x − x x − x x − x 2 − 1 0 − 2 1 − 0

30 Element mass matrix The mass matrix is related to the bilinear form

m(u, v) = c uv dx. ZΩ

The element mass matrix MK of the cell K can be computed as

m (N ,N ) = c(x) N (x)N (x) dx = c(Φ ξ)N (ξ)N (ξ) F dξ. K K,Pj K,Pi K,Pj K,Pi K j i | K | ZK ZKb b b In case of a constant function cK in the cell K we can write

1 6 cK K i = j, (MK )ij = | | 1 c K i = j. ( 12 K | | 6

Element load vector The element load vector is related to the linear form

`K (v) = fvdx. ZK For general, smooth function f we use numerical quadrature to evaluate the integrals. The simplest quadrature rule is

fvdx K f(x )v(x ) ≈ | | K K ZK where xK is the barycenter of the cell K. As this quadrature rule is only exact for linear functions, and the shape functions are already linear, it will provide only reasonable results if f is (almost) constant in K.

3.2.3 Numerical quadrature Numerical quadrature for triangular cells are deﬁned on the triangle with nodes ( 1, 1), (1, 1), ( 1, 1), − − − − where integrals over K can easily transformed to

1 1−ξ1 1 −ξ1 n b 1 1 ξ +1 ξ +1 f(ξ) dξ dξ = f( ξ1+1 , ξ2+1 ) dξ dξ w f( 1,j , 2,j ). 2 1 4 2 2 2 1 ≈ 4 j 2 2 0 0 −1 −1 j=1 Z Z Z Z X There are Gauß quadrature rules for triangles which are exact for polynomials of maximal total degree 1, 2,..., 5 and which use only 1, 3, 4, 6 and 7 points, respectively. See, e. g., page 141 in Sol´ın[ˇ Sol06ˇ ]. To get higher accuracies in a systematic matter there is, for example, the Duﬀy transformation, which transforms an integral over the reference triangle K by an integral over a reference square and employs a tensor-product quadrature rule. After applying the Duﬀy transformation, we integrate over b ξ2 ξ2

b ξ = ξ (1 ξ ) 1 1 − 2 ξ = ξ 2 b2 b b

ξ1 ξ1

Figure 3.6: Duﬀy transformationb of a square into a triangle.

31 the reference cell [0, 1]2, so a tensor product of 1D quadrature rule can be applied. The quadrature rules in 1D are, however, usually given in the interval [ 1, 1]. Hence, we have to transform 1D integrals via − 1 1 1 1 n f(ξ) dξ = f( ξ+1 ) dξ w f( ξj +1 ), 2 2 ≈ 2 j 2 0 −1 j=1 Z Z X where wj are the weights and ξj the abscissas of the quadrature rule. Accurate quadrature rule for smooth functions are variants of the Gauß quadrature. The most well- known is the Gauß-Legendre rule for which the abscissas are the zeros of the n-th Legendre polynomial Pn(ξ) and the weights are given by (see [AS64, page 887])

2 w = . j (1 ξ2)[P 0 (ξ )]2 − j n j The Gauß-Legendre rule is exact for polynomials of degree 2n 1 and the remainder (for the interval [0, 1]) is −

(n!)4 R = f (2n)(ξ), 0 < ξ < 1. n (2n + 1)[(2n)!]3

The zeros of the Legendre polynomials are tabulated (see [AS64, page 921ﬀ]) and there is an algorithm (see Matlab version gauleg.m on the webpage of the lecture) in the numerical recipes [PTFV07]. The Gauß-Lobatto rule is only accurate for polynomials of degree 2n 3. However, both end-points are included in the set of abscissas. − For the square [0, 1]2 we have the product quadrature rule

n n 1 1 1 1 1 1 1 2 f(ξ) dξ dξ = f( ξ1+1 , ξ2+1 ) dξ dξ w w f( ξi+1 , ξj +1 ), 1 2 4 2 2 1 2 ≈ 4 i j 2 2 0 0 −1 −1 i=1 j=1 Z Z Z Z X X with n1 and n2 quadrature points in the two directions. It is important to note that the Duﬀy transformation changes the total polynomial degree. To see this point, let us start with a deﬁnition

Deﬁnition 3.4. Given a domain K Rd, d N, we write ⊂ ∈

α α1 αd d m(K) := span ξ K ξ := ξ ... ξ , α N , α m P { ∈ 7→ 1 · · d ∈ 0 | | ≤ } for the vector space of d-variate polynomials of (total) degree m, m N0. T d ∈ If m = (m1, . . . , md) N we designate by ∈ 0

α1 αd m(K) := span ξ K ξ ... ξ , 0 α m , 1 k d Q { ∈ 7→ 1 · · d ≤ k ≤ k ≤ ≤ } the space of tensor product polynomials of maximal degree mk in the k-th coordinate direction.

Let K = convex 0 , 1 , 1 and K =]0, 1[2. Then { 0 0 1 } b e f(ξ , ξ ) dξ = f(ξ (1 ξ ), ξ ) (1 ξ ) dξ . 1 2 1 − 2 2 − 2 ZKb ZKe e e e e e If f m(K), then the integrand on the right hand side will belong to m,m+1(K), i. e. the degree of polynomial∈ P is increased by one in the second variable. Q The usageb of numerical quadrature inevitably introduces another approximation,e which will contribute to the overall discretization error. The general rule is that

The error due to numerical quadrature must not dominate the total discretization error in the relevant norms.

Remark 3.5. An alternative to numerical quadrature is polynomial interpolation followed by analytical evaluation of the localized integrals.

32 3.2.4 A-priori estimate of the discretization error If the variational problem is elliptic (see Céa’slemma 2.28) the discretization error is bounded “quasi optimally” by the best approximation error b u un V k k inf u wn V . (3.2) k − k ≤ γn wn∈Wn k − k In case of more general bilinear forms that satisfy (IS1), (IS2) and (DIS) we have b u un V 1 + k k inf u wn V . k − k ≤ γ wn∈Wn k − k n For any u V we define some projection I : V W , that we can estimate. Trivially, we have • ∈ n → n inf u wn V u In u V . (3.3) wn∈Wn k − k ≤ k − k For example, for bounded functions u, e. g., for u H2(Ω) we can project onto the space of linear • finite elements by ∈

In u = u(P )bP (x), P ∈NX(M) which is an evaluation of the values on the nodes. The interpolation operator of Raviart-Thomas is deﬁned as • N

(In v)(x) := lj(v)bj(x) j=1 X where lj : V R/C are functionals, called global degrees of freedom. Number of global degrees of freedom equals→ the number of basis functions. Lemma 3.6 (Bramble-Hilbert lemma). If Ω Rd is a bounded Lipschitz-domain and m N, then ⊂ ∈ m γ = γ(m, Ω) > 0 : inf v p Hm(Ω) γ v Hm(Ω) v H (Ω) . ∃ p∈Pm−1(Ω) k − k ≤ | | ∀ ∈ Generalization of Poincaré’s inequality (Lemma 2.26) and a best approximation of polynomials in • Sobolev spaces which is similar to the pointwise statement of Taylor’s theorem. The constant γ depends on the domain. Using reference element K is beneficial as the constant is • then always the same. b The global degrees of freedom, e. g., point evaluations, edge integrals or cell integrals, define on K • local degrees of freedom and the local interpolation operator b Iu = (In u)(x(ξ)),

which acts on the pull-back u(ξ) = u(ΦK (ξ)) of u. bb Estimate on the reference elementb Using the triangle inequality, the fact that there exists a constant C(t, m, K), 1 t m + 1 if d = 1 and 2 t m + 1 if d = 2, 3, such that ≤ ≤ ≤ ≤

Iu t C(t, m, K) u t b k kH (Kb ) ≤ k kH (Kb ) for all u Ht(K), which follows from the continuous embedding of Ht(K) in C0(K) and, ﬁnally, using ∈ bb b b the Bramble-Hilbert lemma we obtain an estimate for the interpolation error of I. Let us measure the error inb the Hr-normb with r t m + 1 (t 1 for d = 1 and t 2 for db= 2, 3). Thenb ≤ ≤ ≥ ≥ b u Iu u Iu − Hr (Kb ) ≤ − Ht(Kb ) = inf (u p) I(u p) b b Ht(Kb ) b b p∈Pb m(Kbb) − − −

inf bu p b b + I (u p) Ht(Kb ) Ht(Kb ) ≤ p∈Pm(Kb ) − − (3.4) b (1 + C(t, m,bK)) inf u bp Ht(Kb ) ≤ p∈Pm(Kb ) k − k

γ(m, K)(1 + bC(t, m, K)) u t . ≤ | b|H (Kb ) =C(t,m), since Kb is ﬁxed b b b | {z } 33 Transformation techniques To apply the Bramble-Hilbert lemma on K and transforming back to K we transform Sobolev norms. b Lemma 3.7. If ΦK : K K is an aﬃne mapping ξ FK ξ + τ , then, for all m N0, → 7→ ∈ m + d m m −1/2 m u m b d FK det(FK ) u m u H (K) , | |H (Kb ) ≤ d k k | | | |H (K) ∀ ∈ m + d m −1 m 1/2 m ub m d F det(F ) u m u H (K) . | |H (K) ≤ d k K k | K | | |H (Kb ) ∀ ∈ with F denoting the matrix norm of FK associated with the Euclidean vector norm.b k kK b Estimate on a cell K With the transformation techniques presented above we can relate the interpolation error on K with that one of the pull-back u r 2 2 u I u = u I u l − n Hr (K) | − n |H b(K) `=0 Xr 2 ` + d 2` 2 d2` F−1 det(F ) u I u ≤ d k K k | K | − n Hl(Kb ) `=0 X Ibub 2 b d r + d 2r |{z} 2 C(r) d2r F−1 det(F ) u Iu . ≤ d k K k | K | − Hr (Kb)

Using the estimate (3.4) on the reference element we get b bb

r + d r −1 r 1/2 u I u C(r, t, m) d F det(F ) u t , − n Hr (K) ≤ d k K k | K | | |H (Kb ) and transformed back to K b

r + d t + d r+t −1 r t u I u C(r, t, m) d F F u t , (3.5) − n Hr (K) ≤ d d k K k k K k | |H (K)

C(d,t,r,m) with r t m + 1, t 2. The estimate| depends{z on the size and} shape of the triangle through FK . ≤ ≤ ≥ Note that convergence in terms of hK is only expected if the solution has higher regularity, i. e., u H2(K) if we measure in H1(K). ∈ Deﬁnition 3.8. Given a cell K of a mesh we deﬁne its diameter M h := sup x y , x, y K , K {| − | ∈ } and the maximum radius of an inscribed ball

r := sup r > 0 : x K : x y < r y K . K { ∃ ∈ | − | ⇒ ∈ } The ratio ρK = hK /rK is called the shape regularity measure of K.

rK l

rK α

Figure 3.7: Diameter hK and maximum radius rK Figure 3.8: Angle condition for shape regularity of of an inscribed ball for a triangular cell. a triangle.

The shape regularity measure of a simplex can be calculated from bounds for the smallest and • largest angles enclosed by edge/face normals. For triangles with the smallest angle α > 0 we have • −1 −1 sin(α/2) ρ 2 sin(α/2) . ≤ K ≤

34 Estimates for aﬃne elements maps

1−d ρM d d −1 −1 d hM det(FK ) hM , FK hM , FK ρMµM hM , (3.6) µM ≤ | | ≤ k k ≤ ≤ with the meshwidth

hM := max h ,K , { K ∈ M} the shape regularity measure

ρM := max ρ ,K , { K ∈ M} and the quasi-uniformity measure

0 −1 µM := max h /h 0 ,K,K = max h ,K min h ,K . { K K ∈ M} { K ∈ M} · { K ∈ M} Interpolation error estimate in the mesh The interpolation error can be decomposed into the contributions from all triangles of , using (3.5) and (3.6), and summing the errors from each triangle we get M

Theorem 3.9. Let In stand for the finite element interpolation operator belonging to the finite element space Sm( ) on a simplicial mesh . Then, for 2 t m + 1, 0 r t M M ≤ ≤ ≤ ≤ t−r t γ = γ(t, r, m, ρM, µM): u I u r γ h u t u H (Ω) . ∃ k − n kH (Ω) ≤ M | |H (Ω) ∀ ∈ Discretization error in the mesh From (3.2) and (3.3), i. e., from the fact that Discretization error best-approximation error interpolation error ≤ ≤ and Theorem 3.9 we can deduce that the discretization error satisfies

t−1 t u u 1 γ h u t for 2 t m + 1 and u H (Ω) , (3.7) k − nkH (Ω) ≤ M | |H (Ω) ≤ ≤ ∈ where γ = γ(Ω, γ , b , ρM, µM, m, t). n k k Discussion To expect convergence we need u to be of higher regularity. For certain problems higher regularity • u Hr+2(Ω) is guaranteed by elliptic shift theorems if f Hr(Ω), and the boundary and the boundary∈ data are smooth. ∈ The bound (3.7) can be interpreted as asymptotic a-priori error estimate • t−1 u un 1 = O(h ) for n , k − kH (Ω) Mn → ∞

where n, n N, is a sequence of simplicial, uniformly shape-regular (ρMn < ρmax for all n N) M ∈ ∈ meshes of Ω, that become infinitely fine (hMn 0 as n ). In other words, we obtain a convergence with the meshwidth of order t 1,→ which is equal→ ∞ to the polynomial degree if the solution u is sufficiently regular. − If Ω is convex or has C1-boundary, and a C1(Ω), then • ∈

u u 2 γhM u u 1 k − nkL (Ω) ≤ k − nkH (Ω)

with γ = γ(Ω, a, m, ρM, µM) > 0. In other words, the weaker the norm of the discretization error the faster it converges to zero as hM 0. → A more sophisticated best approximation estimates for increasing polynomial degree, i. e., • m, t 1, reads ≥ min{m+1,t}−1 hM inf u vn 1 γ(ρM, µM) u t (3.8) m H (Ω) H (Ω) vn∈S (M) k − k ≤ m k k which is an a-priori error estimates for the p-version of H1-conforming ﬁnite elements.