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Chapter 2 Finite Element Methods for Elliptic Boundary Value Problems

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Chapter 2 Finite Element Methods for Elliptic Boundary Value Problems Chapter 2 Finite element methods for elliptic boundary value problems 2.1 Lax-Milgram lemma 2.2 Sobolev spaces In this section, we compile a number of important definitions and results from analysis without proofs. Goal: Construct completions of \classical" function spaces (such as C1(Ω) \ C(Ω)). Proposition 2.2.1 (Completions) If V = (V; h·; ·i) is a vector space with inner product h·; ·i, then there is a unique (up to isometric isomorphisms) Hilbert space (V; hh·; ·ii) such that V is dense in V and hhu; vii = hu; vi for all u; v 2 V ⊆ V: The space (V; hh·; ·ii) is called the completion of (V; h·; ·i). The completion of V is the smallest complete space which contains V . Idea of the proof: Consider equivalence classes of Cauchy sequences in V . (vn)n ∼ (un)n () lim kvn − unk = 0 n!1 Similar to the completion Q −! R. Assumption for the rest of this chapter: Ω ⊂ Rd is a bounded Lipschitz domain with boundary Γ. This means: • Ω is open, bounded, connected, and non-empty. 1 2 Finite element methods (Version: November 2, 2016) • For every boundary point x 2 Γ, there is a neighbourhood U such that, after an affine change of coordinates (translation and/or rotation), Γ \ U is described by the equation xd = χ(x1; : : : ; xd−1) with a uniformly Lipschitz continuous function χ. Moreover, Ω \ U is on one side of Γ \ U. Example: Circles and rectangles are Lipschitz domains, but a circle with a cut is not. Definition 2.2.2 (Lp(Ω) spaces) For p 2 N: Z p Lp(Ω) := v :Ω ! R measurable and jv(x)j dx < 1 Ω 1 Z p p kvkLp(Ω) := jv(x)j dx Ω For p = 1: L1(Ω) := fv :Ω ! R measurable and jv(x)j < 1 a.e.g kvkL1(Ω) := inffc > 0 : jv(x)j ≤ c a.e.g The integral is the Lebesgue integral, and the abbreviation \a.e." means \almost every- where", i.e. for all x 2 Ω n N for null sets N. For convenience, we write kvkLp instead of kvkLp(Ω) if it is clear which domain is meant. The elements of Lp(Ω) are not functions but equivalence classes of functions: u; v 2 Lp(Ω) are equivalent if u = v a.e.. It is common usage to speak of \Lp functions", but some care is required: It does not make sense to speak about \the value of a Lp function in a point x" because a single point is a null set. Important properties: • k·kLp is a norm on Lp(Ω) for every p 2 f1; 2; :::; 1g, and (Lp(Ω); k·kLp ) is a Banach space (complete). • Special case p = 2: The space L2(Ω) is a Hilbert space with inner product Z hu; vi = u(x)v(x) dx: L2(Ω) Ω Definition 2.2.3 loc n o L1 (Ω) := u :Ω −! R is measurable and locally integrable \Locally integrable" means that u 2 L1(S) for all compact subsets S ⊂ Ω. Tobias Jahnke 3 loc Definition 2.2.4 (Weak derivative) A function u 2 L1 (Ω) is weakly differen- loc tiable with respect to xi if there is a v 2 L1 (Ω) such that Z Z 1 v(x)φ(x) dx = − u(x)@xi φ(x) dx for all φ 2 Cc (Ω) Ω Ω or equivalently hv; φi = − hu; @ φi for all φ 2 C1(Ω): L2(Ω) xi L2(Ω) c Then, v is called the weak (partial) derivative and is denoted by v = @xi u. Uniqueness: If v = @ u 2 Lloc(Ω) and v = @ u 2 Lloc(Ω) are both weak partial xi 1 e xi 1 loc derivatives of u 2 L1 (Ω), then v = ve a.e. Consistency of the definition: If u 2 C1(Ω)\C(Ω), then the weak derivative coincides with the classical derivative (exercise). d Notation: For a multi-index α = (α1; : : : ; αd) 2 N0 we set @αu := @α1 :::@αd u; jαj = α + ::: + α : x1 xd 1 1 d The above definition can now be extended to higher-order weak derivatives: v(x) = @αu(x) is the weak partial derivative of u(x) if Z Z jαj1 α 1 v(x)φ(x) dx = (−1) u(x)@ φ(x) dx for all φ 2 Cc (Ω): Ω Ω Definition 2.2.5 (Sobolev space Hm(Ω)) For m 2 N the Sobolev space Hm(Ω) is the set m n α d o H (Ω) := v 2 L2(Ω) : @ v 2 L2(Ω) exists for all α 2 N0 with jαj1 ≤ m with inner product X α α hu; vi m := h@ u; @ vi : H (Ω) L2(Ω) jαj1≤m q and norm kvkHm(Ω) = hv; viHm(Ω). As for the Lp spaces, we often write hu; viHm and k · kHm instead of hu; viHm(Ω) and k · kHm(Ω): Example (m = 1): 1 n o H (Ω) := v 2 L2(Ω) : @xi v 2 L2(Ω) exists for all i = 1; :::; d d X hu; vi 1 := hu; vi + h@ u; @ vi H (Ω) L2(Ω) xi xi L2(Ω) i=1 d !1=2 2 X 2 1 kvkH (Ω) = kvkL2(Ω) + k@xi vkL2(Ω) : i=1 4 Finite element methods (Version: November 2, 2016) Important properties: m 0 H (Ω); h·; ·iHm is a Hilbert space for every m 2 N0. (We define H (Ω) = L2(Ω).) In fact, the space Z 1;m n 1 α 2 d o C (Ω) := v 2 C (Ω) : j@ v(x)j dx < 1 for all α 2 N0 with jαj1 ≤ m : Ω m m is dense in H (Ω) with respect to k · kHm : For every u 2 H (Ω) and every " > 0 there 1;m is a v" 2 C (Ω) such that kv" − ukHm < " . m 1;m It can be shown that H (Ω) is the completion of C (Ω) with respect to k · kHm , i.e. m 1;m u 2 H (Ω) () There are vn 2 C (Ω) such that lim ku − vnkHm(Ω) = 0: n!1 Example: Let Ω = (−1; 1) and u(x) = jxj. Then u 2 H1(Ω), but u 62 C1;1(Ω). However, q 2 1 1;1 we can define vn := x + n2 2 C (Ω) and show that kvn − ukH1(Ω) −! 0 for n −! 1 (exercise). m 1 Definition 2.2.6 The Sobolev space H0 (Ω) is the completion of Cc (Ω) with respect to k · kHm(Ω), i.e. m 1 u 2 H0 (Ω) () There are vn 2 Cc (Ω) such that lim ku − vnkHm(Ω) = 0: n!1 m m 1 m H0 (Ω) is a closed subspace of H (Ω). If the boundary Γ is C , then v 2 C(Ω) \ H0 (Ω) implies that v(x) = 0 for all x 2 Γ. Proposition 2.2.7 (Poincar´e-Friedrichs inequality) Let 1 d ! 2 X 2 1 jvjH (Ω) := k@xi vkL2(Ω) i=1 denote the Sobolev seminorm. There is a constant cΩ > 0 such that 1 kvkH1(Ω) ≤ cΩjvjH1(Ω) for all v 2 H0 (Ω): Proof. See 1.5, page 29 in [Bra07]. 1 Remark: On H0 (Ω), the seminorm j · jH1(Ω) is even a norm and j · jH1(Ω) ∼ k · kH1(Ω) (exercise). If we want to solve a PDE with inhomogeneous boundary conditions, we need a condition such as, e.g., \u(x) = g(x) for all x 2 Γ”. Since Γ is a null set, however, we have to specify how such a condition has to be understood if u 2 L2(Ω). Tobias Jahnke 5 Theorem 2.2.8 (Trace theorem) There is a bounded linear map 1 1 γ : H (Ω) −! L2(Γ); kγ(v)kL2(Γ) ≤ ckvkH (Ω) (c > 0) 1 ¯ such that γ(v) = vjΓ for all v 2 C (Ω) Proof: page 45-47 in [Bra07] Remark: It can be shown that 1 1 H0 (Ω) := v 2 H (Ω) : γ(v) = 0 d Theorem 2.2.9 (Sobolev's embedding theorem) Let Ω ⊂ R and let r; s 2 N0 with r > s + d=2. Then Hr(Ω) ⊂ Cs(Ω) and there is a constant c > 0 such that X α kukCs(Ω) ≤ ckukHr(Ω); where kukCs(Ω) := sup j@ u(x)j: x2Ω jαj1≤s Examples: d = 1 =) H1(Ω) ⊂ C(Ω);H2(Ω) ⊂ C1(Ω) d 2 f2; 3g =) H1(Ω) 6⊂ C(Ω);H2(Ω) ⊂ C(Ω) 2.3 Weak solutions of elliptic boundary value prob- lems Let Ω ⊆ Rd be a bounded Lipschitz domain with d 2 f2; 3g and assume that Γ is piecewise C1. Consider the elliptic PDE Au = f for f 2 L2(Ω) and with the second-order differential operator Au(x) = −div κ(x)ru(x) + κ0(x)u(x) d X = − @xi κij(x)@xj u(x) + κ0(x)u(x) i;j=1 d×d with κ(x) = κij(x) i;j 2 R for all x 2 Ω. The coefficient functions κij :Ω −! R and κ0 :Ω −! R are supposed to have the following properties: 1. jκ0(x)j; jκij(x)j ≤ M for all x 2 Ω 6 Finite element methods (Version: November 2, 2016) 2. κij = κji for all i; j = 1; :::; d 3. There are constants c0 ≥ 0 and c1 > 0 such that κ0(x) ≥ c0 and d d d X X X 2 T d κij(x)ξiξj ≥ c1 ξi for all x 2 Ω and ξ = (ξ1; : : : ; ξd) 2 R : (2.1) i=1 j=1 i=1 The second assertion is equivalent to T 2 T ξ κξ ≥ c1kξk2; ξ = (ξ1; :::; ξd) : A differential operator with the properties2 and3 is called elliptic.
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