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

The Maximum Principle Theorem Suppose L h U j = � i ∈ J \{ j } c ij U i − c j U j , ∀ j ∈ J Ω ; J and L h satisfy (1) J D � = ∅ , and J is J D connected with respect to L h ; � (2) c j > 0 , c ij > 0 , ∀ i ∈ D L h ( j ) , and c j ≥ c ij . i ∈ D Lh ( j ) Suppose the grid function U satisfies L h U j ≥ 0 , ∀ j ∈ J Ω . Then, � � M Ω � max U i ≤ max max U i , 0 . i ∈ J Ω i ∈ J D Furthermore, if J and L h satisfy (3) : J is connected with respect to L h ; and there exists interior node j ∈ J Ω such that U j = max i ∈ J U i ≥ 0 . Then, U must be a constant on J.

Finite Difference Methods for Elliptic Equations Error Analysis Based on the Maximum Principle The Maximum Principle The Existence Theorem Theorem Suppose the grid J and the linear operator L h satisfy the conditions (1) and (2) of the maximum principle. Then, the difference equation � − L h U j = f j , ∀ j ∈ J Ω , U j = g j , ∀ j ∈ J D , has a unique solution. proof: We only need to show that L h U j = 0 , ∀ j ∈ J Ω ; U j = 0 , ∀ j ∈ J D ⇒ U j = 0 , ∀ j ∈ J . In fact, by the maximum principle L h U ≥ 0 implies U ≤ 0, and by the corollary of the maximum principle, L h U ≤ 0 implies U ≥ 0, � thus U ≡ 0 on J . 3 / 38

Finite Difference Methods for Elliptic Equations Error Analysis Based on the Maximum Principle The Maximum Principle ( − L h ) − 1 is a Positive Operator Corollary Suppose the grid J and the linear operator L h satisfy the conditions (1) and (2) of the maximum principle. Then, f j ≥ 0 , ∀ j ∈ J Ω , g j ≥ 0 , ∀ j ∈ J D , ⇒ U j ≥ 0 , ∀ j ∈ J ; and f j ≤ 0 , ∀ j ∈ J Ω , g j ≤ 0 , ∀ j ∈ J D , ⇒ U j ≤ 0 , ∀ j ∈ J ; The corollary says that ( − L h ) − 1 is a positive operator, i.e. ( − L h ) − 1 ≥ 0. In other words, every element of the matrix ( − L h ) − 1 is nonnegative. In fact, the matrix − L h is a M matrix, i.e. the diagonal elements of A are all positive, the off-diagonal elements are all nonpositive, and elements of A − 1 are all nonnegative. 4 / 38

Finite Difference Methods for Elliptic Equations Error Analysis Based on the Maximum Principle The Comparison Theorem and the Stability The Comparison Theorem and the Stability Theorem Suppose the grid J and the linear operator L h satisfy the conditions (1) and (2) of the maximum principle. Let the grid function U be the solution to the linear difference equation � − L h U j = f j , ∀ j ∈ J Ω , U j = g j , ∀ j ∈ J D . Let Φ be a nonnegative grid function defined on J satisfying L h Φ j ≥ 1 , ∀ j ∈ J Ω . | U j | ≤ max | U j | + max | f j | . max Φ j max Then, we have j ∈ J Ω j ∈ J D j ∈ J D j ∈ J Ω 5 / 38

Finite Difference Methods for Elliptic Equations Error Analysis Based on the Maximum Principle The Comparison Theorem and the Stability The Proof of the Comparison Theorem Proof: Firstly, it follows from the maximum principle that 0 ≤ max Φ j ≤ max Φ j . j ∈ J Ω j ∈ J D Next, define � � Ψ ± j = ± U j + max | f i | Φ j , ∀ j ∈ J . i ∈ J Ω It is easily verified that L h Ψ ± j ≥ 0 on J Ω , thus by the maximum principle ± U j ≤ Ψ ± j ≤ max | U j | + max | f j | , ∀ j ∈ J Ω , Φ j max j ∈ J D j ∈ J D j ∈ J Ω � since Φ is nonnegative. 6 / 38

Finite Difference Methods for Elliptic Equations Error Analysis Based on the Maximum Principle The Comparison Theorem and the Stability The Comparison Theorem and the Stability If the grid J and the linear operator L h satisfy the conditions (1) and (2) of the maximum principle. Let the grid function U be the solution to the linear difference equation � − L h U j = f j , ∀ j ∈ J Ω , U j = g j , ∀ j ∈ J D . Then, the comparison theorem says that | U j | ≤ max | g j | + max | f j | , max Φ j max j ∈ J Ω j ∈ J D j ∈ J D j ∈ J Ω in other words, the finite difference scheme is stable in the L ∞ norm � · � ∞ , as long as there is a nonnegative Φ s.t. L h Φ ≥ 1, and max j ∈ J D Φ j is uniformly bounded with respect to J . 7 / 38

A Priori Error Estimate Theorem Suppose the grid J and the linear operator L h satisfy the conditions (1) and (2) of the maximum principle. Let Φ be a nonnegative grid function defined on J satisfying L h Φ j ≥ 1 , ∀ j ∈ J Ω . Then, the error of the finite difference approximation equation e h = { U j − u j } j ∈ J Ω can be bounded by the error on the Dirichlet boundary and the local truncation error T h = { L h ( U j − u j ) } j ∈ J Ω in the sense that max | e j | ≤ max | e j | + max Φ j max | T j | . j ∈ J Ω j ∈ J D j ∈ J D j ∈ J Ω T j can be of different order on the regular and irregular interior nodes. Is it possible for us to choose different comparison functions for them, so that better error estimates can be obtained?

A Generalized version of the Comparison Theorem Theorem Suppose the grid J and the linear operator L h satisfy the conditions (1) and (2) of the maximum principle. Let the grid function U be the solution to the linear difference equation � − L h U j = f j , ∀ j ∈ J Ω , U j = g j , ∀ j ∈ J D . Let Φ be a nonnegative grid function defined on J satisfying � L h Φ j ≥ C 1 > 0 , ∀ j ∈ J Ω 1 , L h Φ j ≥ C 2 > 0 , ∀ j ∈ J Ω 2 , where J Ω 1 ∪ J Ω 2 = J Ω , J Ω 1 ∩ J Ω 2 = ∅ . Then � � C − 1 | f j | , C − 1 max | U j | ≤ max | U j | + max Φ j max max max | f j | . 1 2 j ∈ J Ω j ∈ J D j ∈ J D j ∈ J Ω1 j ∈ J Ω2

Finite Difference Methods for Elliptic Equations Error Analysis Based on the Maximum Principle Comparison Theorem and A Priori Error Estimation Proof of the Generalized Comparison Theorem Proof: Firstly, it follows from the maximum principle that 0 ≤ max Φ j ≤ max Φ j . j ∈ J Ω j ∈ J D Next, define � � Ψ ± C − 1 | f j | , C − 1 j = ± U j + max max max | f j | Φ j , ∀ j ∈ J . 1 2 j ∈ J Ω1 j ∈ J Ω2 We have L h Ψ ± j ≥ 0 on J Ω , thus, by the maximum principle, � � ± U j ≤ Ψ ± C − 1 | f j | , C − 1 j ≤ max | U j | + max Φ j max max max | f j | , 1 2 j ∈ J D j ∈ J D j ∈ J Ω1 j ∈ J Ω2 � since Φ is nonnegative. 10 / 38

A Better a Priori Error Estimate Theorem Suppose the grid J and the linear operator L h satisfy the conditions (1) and (2) of the maximum principle. Let Φ be a nonnegative grid function defined on J satisfying � L h Φ j ≥ C 1 > 0 , ∀ j ∈ J Ω 1 , L h Φ j ≥ C 2 > 0 , ∀ j ∈ J Ω 2 , Then, the error of the finite difference approximation equation e h = { U j − u j } j ∈ J Ω satisfies � � C − 1 | T j | , C − 1 max | e j | ≤ max | e j | + max Φ j max max max | T j | . 1 2 j ∈ J Ω j ∈ J D j ∈ J D j ∈ J Ω1 j ∈ J Ω2 We will see that, by defining proper Φ, this can actually produce ”optimal” error estimate for Dirichlet boundary value problems of elliptic equations defined on domains with curved boundaries.

Finite Difference Methods for Elliptic Equations Error Analysis Based on the Maximum Principle Comparison Theorem and A Priori Error Estimation An Example of Optimal Error Estimate 1 Poisson equation defined on 2-D region with curved boundary; 2 Uniform grid with h x = h y = h ; ◦ 3 On J Ω =: J Ω 1 , L h is the standard 5-point difference scheme; ◦ 4 On J Ω \ J Ω =: J Ω 2 , L h is a symmetric 5-point scheme on nonuniform grid, for example (see (1.3.24)) L h U P = 1 � U E ∗ − U P − U P − U W � + 1 � U N ∗ − U P − U P − U S � . h x h ∗ h x h y h ∗ h y x y 5 Define ˜ J D = J D ∩ [ ∪ j ∈ J Ω2 D L h ( j )]. 12 / 38

Finite Difference Methods for Elliptic Equations Error Analysis Based on the Maximum Principle Comparison Theorem and A Priori Error Estimation An Example of Optimal Error Estimate 6 The local truncation error satisfies | T j | ≤ K 1 h 2 , max max | T j | ≤ K 2 , j ∈ J Ω1 j ∈ J Ω2 where K 1 and K 2 are constants independent of h . 7 (¯ x , ¯ y ) is the circumcenter of Ω, R is the circumradius. 8 Take comparison functions of the following form � x ) 2 + ( y − ¯ ∀ ( x , y ) �∈ ˜ � y ) 2 � Φ( x , y ) = E 1 ( x − ¯ , J D , x ) 2 + ( y − ¯ ∀ ( x , y ) ∈ ˜ � y ) 2 � Φ( x , y ) = E 1 ( x − ¯ + E 2 , J D , where E 1 and E 2 are positive undetermined coefficients; 13 / 38

Finite Difference Methods for Elliptic Equations Error Analysis Based on the Maximum Principle Comparison Theorem and A Priori Error Estimation An Example of Optimal Error Estimate 9 Since D L h ( j ) ∩ ˜ J D � = ∅ if and only if j ∈ J Ω 2 , we have 0 ≤ Φ j ≤ E 1 R 2 + E 2 ,  ∀ j ∈ J D ,   L h Φ j = 4 E 1 , ∀ j ∈ J Ω 1 , L h Φ j ≥ E 1 + E 2 h − 2 ≥ E 2 h − 2 ,  ∀ j ∈ J Ω 2 .  The last inequality follows from h ∗ x + h x ≥ 1 1 ≥ h − 2 and 2, 2 h x h x h ∗ x x ) 2 = L h ( y − ¯ y ) 2 = 2. L h ( x − ¯ 14 / 38