Numerical Solutions to Partial Differential Equations Zhiping Li - PowerPoint PPT Presentation

Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University

Error Estimates of Finite Element Solutions C´ ea Lemma and Abstract Error Estimates C´ ea Lemma C´ ea Lemma — an Abstract Error Estimate Theorem 1 Consider the variational problem of the form � Find u ∈ V such that a ( u , v ) = f ( v ) , ∀ v ∈ V . 2 Consider the conforming finite element method of the form � Find u h ∈ V h ⊂ V such that a ( u h , v h ) = f ( v h ) , ∀ v h ∈ V h . 3 The problem: how to estimate the error � u − u h � ? 4 The method used for FDM is not an ideal framework for FEM. 5 The standard approach for the error estimations of a finite element solution is to use an abstract error estimate to reduce the problem to a function approximation problem. 2 / 22

Error Estimates of Finite Element Solutions C´ ea Lemma and Abstract Error Estimates C´ ea Lemma C´ ea Lemma — an Abstract Error Estimate Theorem Theorem Let V be a Hilbert space, V h be a linear subspace of V . Let the bilinear form a ( · , · ) and the linear form f ( · ) satisfy the conditions of the Lax-Milgram lemma (see Theorem 5.1) . Let u ∈ V be the solution to the variational problem, and u h ∈ V h satisfy the equation a ( u h , v h ) = f ( v h ) , ∀ v h ∈ V h . Then, there exist a constant C independent of V h , such that � u − u h � ≤ C inf v h ∈ V h � u − v h � , where � · � is the norm of V . 3 / 22

Error Estimates of Finite Element Solutions C´ ea Lemma and Abstract Error Estimates C´ ea Lemma Proof of the C´ ea Lemma 1 Since u and u h satisfy the equations, and V h ⊂ V , we have a ( u − u h , w h ) = a ( u , w h ) − a ( u h , w h ) = f ( w h ) − f ( w h ) = 0 , ∀ w h ∈ V h . 2 In particular, taking w h = u h − v h leads to a ( u − u h , u h − v h ) = 0 . α � u − u h � 2 ≤ a ( u − u h , u − u h ). 3 The V -ellipticity ⇒ 4 The boundedness ⇒ a ( u − u h , u − v h ) ≤ M � u − u h �� u − v h � . 5 Hence, α � u − u h � 2 ≤ a ( u − u h , u − v h ) ≤ M � u − u h �� u − v h � . 6 Take C = M /α , we have � u − u h � ≤ C � u − v h � , ∀ v h ∈ V h . 7 The conclusion of the theorem follows. � 4 / 22

Error Estimates of Finite Element Solutions C´ ea Lemma and Abstract Error Estimates C´ ea Lemma Remarks on the C´ ea Lemma 1 The C´ ea lemma reduces the error estimation problem of � u − u h � to the optimal approximation problem of inf v h ∈ V h � u − v h � . 2 Error of the finite element solution � u − u h � is of the same order as the optimal approximation error inf v h ∈ V h � u − v h � . 3 Suppose the V h -interpolation function Π h u of u is well defined in the finite element function space V h , then, � u − u h � ≤ C inf v h ∈ V h � u − v h � ≤ C � u − Π h u � . 4 Therefore, the error estimation problem of � u − u h � can be further reduced to the error estimation problem for the V h -interpolation error � u − Π h u � . 5 / 22

Error Estimates of Finite Element Solutions C´ ea Lemma and Abstract Error Estimates Geometric Explanation of the C´ ea Lemma For Symmetric a ( · , · ), u h Is a Orthogonal Projection of u on V h 1 If the V -elliptic bounded bilinear form a ( · , · ) is symmetric, then, a ( · , · ) defines an inner product on V , with the induced norm equivalent to the V -norm. 2 Denote P h : V → V h as the orthogonal projection operator induced by the inner product a ( · , · ). Then, a ( u − P h u , v h ) = 0 , ∀ v h ∈ V h . 3 Therefore, the finite element solution u h = P h u , i.e. it is the orthogonal projection of u on V h with respect to the inner product a ( · , · ). 6 / 22

Error Estimates of Finite Element Solutions C´ ea Lemma and Abstract Error Estimates Geometric Explanation of the C´ ea Lemma C´ ea Lemma for Symmetric a ( · , · ) Corollary Under the conditions of the C´ ea Lemma, if the bilinear form a ( · , · ) is in addition symmetric, then, the solution u h is the orthogonal projection, which is induced by the inner product a ( · , · ) , of the solution u on the subspace V h , meaning u h = P h u. Furthermore, we have a ( u − u h , u − u h ) = inf v h ∈ V h a ( u − v h , u − v h ) . The proof follows the same lines as the proof of the C´ ea lemma. The only difference here is that α = M = 1. 7 / 22

Error Estimates of Finite Element Solutions C´ ea Lemma and Abstract Error Estimates Geometric Explanation of the C´ ea Lemma C´ ea Lemma in the Form of Orthogonal Projection Error Estimate Denote ˜ P h : V → V h as the orthogonal projection operator induced by the inner product ( · , · ) V of V , then, � u − ˜ P h u � = � ( I − ˜ P h ) u � = inf v h ∈ V h � u − v h � . Therefore, as a corollary of the C´ ea lemma, we have Corollary Let V be a Hilbert space, and V h be a linear subspace of V . Let a ( · , · ) be a symmetric bilinear form on V satisfying the conditions of the Lax-Milgram lemma. Let P h and ˜ P h be the orthogonal projection operators from V to V h induced by the inner products a ( · , · ) and ( · , · ) V respectively. Then, we have P h � ≤ � I − P h � ≤ M � I − ˜ α � I − ˜ P h � . 8 / 22

Error Estimates of Finite Element Solutions The Interpolation Theory of Sobolev Spaces An Example on Interpolation Error Estimates 1-D Example on Linear Interpolation Error Estimation for H 2 Functions ˆ Ω = (0 , 1), Ω = ( b , b + h ), h > 0. 1 2 F : ˆ x ∈ [0 , 1] → [ b , b + h ], F (ˆ x ) = h ˆ x + b : an invertible affine mapping from ˆ Ω to Ω. ˆ Π : C ([0 , 1]) → P 1 ([0 , 1]): the interpolation operator with 3 ˆ v (0), ˆ Πˆ v (0) = ˆ Πˆ v (1) = ˆ v (1). 4 Π : C ([ b , b + h ]) → P 1 ([ b , b + h ]): the interpolation operator with Π v ( b ) = v ( b ), Π v ( b + h ) = v ( b + h ). 9 / 22

Error Estimates of Finite Element Solutions The Interpolation Theory of Sobolev Spaces An Example on Interpolation Error Estimates 1-D Example on Linear Interpolation Error Estimation for H 2 Functions 5 Let u ∈ H 2 (Ω), denote ˆ u (ˆ x ) = u ◦ F (ˆ x ) = u ( h ˆ x + b ), then, it u ∈ H 2 (ˆ can be shown ˆ Ω), thus, ˆ u ∈ C ([0 , 1]). Π is P 1 ([0 , 1]) invariant : ˆ ˆ Πˆ w = ˆ w , ∀ ˆ w ∈ P 1 ([0 , 1]), thus, 6 � ( I − ˆ Ω = � ( I − ˆ Ω ≤ � I − ˆ Π)ˆ u � 0 , ˆ Π)(ˆ u + ˆ w ) � 0 , ˆ Π � � ˆ u + ˆ w � 2 , ˆ Ω , where � I − ˆ Π � is the norm of I − ˆ Π : H 2 (ˆ Ω) → L 2 (ˆ Ω). ⋆ This shows that I − ˆ Π ∈ L ( H 2 (0 , 1) / P 1 ([0 , 1]); L 2 (0 , 1)), and u − ˆ Ω ≤ � I − ˆ (1) � ˆ Πˆ u � 0 , ˆ Π � inf � ˆ u + ˆ w � 2 , ˆ Ω , w ∈ P 1 (ˆ ˆ Ω) where inf ˆ Ω) � ˆ u + ˆ w � 2 , ˆ Ω is the norm of ˆ u in the quotient space w ∈ P 1 (ˆ H 2 (0 , 1) / P 1 ([0 , 1]). 10 / 22

Error Estimates of Finite Element Solutions The Interpolation Theory of Sobolev Spaces An Example on Interpolation Error Estimates 1-D Example on Linear Interpolation Error Estimation for H 2 Functions ⋆ It can be shown that, ∃ const. C (ˆ Ω) > 0 s.t. Ω ≤ C (ˆ (2) | ˆ u | 2 , ˆ Ω ≤ inf � ˆ u + ˆ w � 2 , ˆ Ω) | ˆ u | 2 , ˆ Ω . w ∈ P 1 (ˆ ˆ Ω) x ) = h 2 u ′′ ( x ). u ′′ (ˆ ⋆ It follows from the chain rule that ˆ ⋆ By a change of the integral variable, and dx = hd ˆ x , we obtain u ∈ H 2 (ˆ u | 2 Ω = h 3 | u | 2 (3) ˆ Ω), and | ˆ 2 , Ω ; 2 , ˆ u − ˆ (4) � u − Π u � 2 u � 2 0 , Ω = h � ˆ Πˆ Ω . 0 , ˆ 11 / 22

Error Estimates of Finite Element Solutions The Interpolation Theory of Sobolev Spaces An Example on Interpolation Error Estimates 1-D Example on Linear Interpolation Error Estimation for H 2 Functions The conclusion (1) says that the L 2 norm of the error of a P 1 invariant interpolation can be bounded by the quotient norm of the function in H 2 (0 , 1) / P 1 ([0 , 1]). The conclusion (2) says that the semi norm | · | 2 , Ω is an equivalent norm of the quotient space H 2 (0 , 1) / P 1 ([0 , 1]). The conclusions (3) and (4) present the relations between the semi-norms of Sobolev spaces defined on affine-equivalent open sets. 12 / 22

Error Estimates of Finite Element Solutions The Interpolation Theory of Sobolev Spaces An Example on Interpolation Error Estimates 1-D Example on Linear Interpolation Error Estimation for H 2 Functions ⋆ The combination of (4) and (1) yields 1 2 � I − ˆ � u − Π u � 0 , Ω ≤ h Π � inf � ˆ u + ˆ w � 2 , ˆ Ω w ∈ P 1 (ˆ ˆ Ω) ⋆ This together with (2) and (3) lead to the expected interpolation error estimate: � u − Π u � 0 , Ω ≤ � I − ˆ Π � C (ˆ Ω) | u | 2 , Ω h 2 , ∀ u ∈ H 2 (Ω) . 13 / 22

Error Estimates of Finite Element Solutions The Interpolation Theory of Sobolev Spaces An Example on Interpolation Error Estimates A Framework for Interpolation Error Estimation of Affine Equivalent FEs 1 The polynomial quotient spaces of a Sobolev space and their equivalent quotient norms ((2) in the example); 2 The relations between the semi-norms of Sobolev spaces defined on affine-equivalent open sets ((3), (4) in the exmample); 3 The abstract error estimates for the polynomial invariant operators ((1) in the example); 4 To estimate the constants appeared in the relations of the Sobolev semi-norms by means of the geometric parameters of the corresponding affine-equivalent open sets. 14 / 22