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Finite Elements 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 0x1 x x = ; x = y ; y @ A z respectively. The Euclidean norm on Rd is denoted as v u d uX 2 jxj = t xi : i=1 3 1.1.2 Notation: Partial derivatives of a function u 2 C1(Rd) are de- noted by @u(x) @ = @x u(x) = @xi u(x) = @iu(x): @xi i The gradient of u 2 C1 is the row vector ru = (@1u; : : : ; @du) The Laplacian of a function u 2 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 inde- pendent variable x. Therefore, ∆u ≡ ∆u(x): 1.1.4 Definition: A linear PDE of second order in divergence form for a function u 2 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 typi- cally 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 2 @Ω, we associate the outer unit normal vector n ≡ n(x). The symbol @nu ≡ (ru)n denotes the normal derivative of a function u 2 C1(Ω) at a point x 2 @Ω. 1.1.8 Definition: We distinguish three types of boundary conditions for Poisson’s equation, namely for a point x 2 @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 2 C2(Ω) \ C(Ω), such that −∆u(x) = f(x) x 2 Ω; (1.3a) u(x) = g(x) x 2 @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 2 C2(Ω) \ C(Ω) solves the Dirichlet problem, then it minimizes the Dirichlet en- ergy Z Z 1 2 E(v) = 2 jrvj dx − fv dx; (1.4) Ω Ω among all functions v from the set 2 Vg = v 2 C (Ω) \ C(Ω) vj@Ω = g : (1.5) This minimizer is unique. Proof. Using variation of E, we will show that d E(u + "v) = 0 d" "=0 for all v 2 V0 since this implies u + "v = g on @Ω. By evaluating the square we have d Z E(u + "v) = rurv + "jrvj2 − fv dx d" Ω Since we are intersted in E(u), we now consider " = 0. We get that u minimizes E(u + "v) at " = 0 implies Z Z rurv dx = fv dx; 8v 2 V0: Ω Ω By Green’s formula Z Z Z rurv dx = −∆uv dx + @nuv ds Ω Ω @Ω we obtain that if u minimizes E(·), then Z Z rurv dx = fv dx; 8v 2 V0; Ω Ω since v 2 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 2 V0. Using Ω fv dx = Ω rurv yields 1 Z 1 E(u+v)−E(u) = jr(u+v)j2−2r(u+v)ru+jruj2 dx = jrvj2 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 fu gn2N converging to this infinum, i.e. lim E(u(n)) = inf E(v): n!1 v2V0 (n) Second, we show that fu gn is a Cauchy sequence. For the first part we use Friedrich’s inequality kvkL2(Ω) ≤ λ(Ω)krvkL2(Ω) v 2 V0: The proof of this result will be given later. Using Hölder’s inequality we obtain Z 1 2 1 2 E(v) = krvkL2(Ω − fv dx ≥ krvkL2(Ω − kfkL2(ΩkvkL2(Ω 2 Ω 2 Applying Friedrich’s inquality yields that the above expression is greater or equal than 1 2 1 krvk 2 − krvk 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 krvk 2 − krvk 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 jv + wj2 + jv − wj2 = 2jvj2 + 2jwj2. Let m; n be natural numbers, then (n) (m) 2 (n) 2 (m) 2 1 (n) (m) 2 ju − u j1 =2ju j1 + 2ju j1 − 4j =2(u + u j1 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 ! 1 yields 4E(u ) + 4E(u ) ! 8 infv2V0 E(v). 1 (n) (m) Lastly, −E( =2f(u + u )) can be bounded by infv2V0 E(v). It follows that (n) (m) 2 lim supm;n!1ju − u j1 ≤ 0 and consequently as desired (n) (m) 2 lim ju − u j1 = 0: m;n!1 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 2 Vg minimizes the Dirichlet energy, if and only if there holds Z Z ru · rv dx = fv dx; 8v 2 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 2 Vg minimizes the Dirichlet energy admits the representation u = ug + u0, where ug 2 Vg is arbitrary and u0 2 V0 solves Z Z Z ru · rv dx = fv dx − rug · rv dx; 8v 2 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 2 V0 there holds kvkΩ ≤ diam(Ω)krvkΩ: (1.8) 8 1.1.18 Lemma: The definitions jvj1 = krvkL2(Ω); (1.9) q 2 2 kvk1 = kvkL2(Ω) + jvj1; both define a norm on V0.
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