Numerical Solutions to Partial Differential Equations

Transcription

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

2 The Residual and Error of Finite Element Solutions Mixed BVP of Poisson Equation on Polygonal Region in R 2 Consider the boundary value problem of the Poisson equation u = f, x Ω, u u = 0, x Ω 0, ν = g, x Ω 1, where Ω is a polygonal region in R 2, Ω 0 is a relative closed subset in Ω with positive 1-dimensional measure, Ω = Ω 0 Ω 1, Ω 0 Ω 1 =, f L 2 (Ω), g L 2 ( Ω 1 ). 2 / 37

3 The Residual and Error of Finite Element Solutions Mixed BVP of Poisson Equation on Polygonal Region in R 2 consider the standard weak form of the problem: { Find u V such that Ω u v dx = Ω f v dx + Ω 1 g v ds, v V, where V = { v H 1 (Ω) : v Ω0 = 0 } ; consider the conforming finite element method based on a family of regular class C 0 type (1) Lagrange triangular elements. 3 / 37

4 A Theorem on the Relation of Residual and Error of a FE Solution 1 Define the residual operator R : V V of the problem by R(v)(w) = f w dx+ g w ds v w dx, w V. Ω Ω 1 Ω 2 The dual norm of the residual of a finite element solution u h : { } R(u h ) V = sup fw dx + gw ds u h w dx w V Ω Ω 1 Ω w 1,2,Ω =1 Theorem Let u V, u h V h be the weak solution and the finite element solution of the problem respectively. Then, there exists a constant C(Ω), which depends only on Ω, such that R(u h ) V u u h 1,2,Ω C(Ω) R(u h ) V.

5 Remarks on Residual Dual Norm Estimation 1 We hope to develop a formula, which is easily computed and involves only available data such as f, g, u h and geometric parameters of the triangulation and thus is usually called an a posteriori error estimator, to evaluate the dual norm of the residual. 2 Recall that in the a priori error estimates, the polynomial invariant interpolation operator plays an important role. For example, write w as (w Π h w) + Π h w can have some advantage. 3 However, the Lagrange nodal type interpolation operators require the function to be at least in C 0. 4 Here, we need to introduce a polynomial invariant interpolation operator for functions in H 1.

6 The Residual and Error of Finite Element Solutions Notations on a Family of Regular Triangular Triangulations {T h (Ω)} h>0 1 E(), N (): the sets of all edges and vertices of T h (Ω). 2 Denote E h := T h (Ω) E(), N h := T h (Ω) N (). 3 N (E): the sets of all vertices of an edge E E h. 4 E h,i := {E E h : E Ω i }, N h,i := N h Ω i, i = 0, 1. 5 E h,ω = E h \ (E h,0 E h,1 ), N h,ω = N h \ (N h,0 N h,1 ). 6 / 37

7 The Residual and Error of Finite Element Solutions Notations on a Family of Regular Triangular Triangulations {T h (Ω)} h>0 6 ω :=, ω E :=, ω x :=. E() E( ) E E( ) x N ( ) 7 ω := N () N ( ), ω E := N (E) N ( ). 8 The corresponding finite element function space: V h = {v C( Ω) : v P 1 (), T h (Ω), v(x) = 0, x N h,0 }. 7 / 37

8 The Residual and Error of Finite Element Solutions The Clément Interpolation Operator I h : V V h Definition For any v V and x N h, denote π x v as the L 2 (ω x ) projection of v on P 1 (ω x ), meaning π x v P 1 (ω x ) satisfies v p dx = ω x (π x v) p dx, ω x p P 1 (ω x ). The Clément interpolation operator I h : V V h is defined by I h v(x) = (π x v)(x), x N h,ω N h,1 ; I h v(x) = 0, x N h,0. 8 / 37

9 The Residual and Error of Finite Element Solutions The Clément Interpolation Operator I h : V V h 1 The Clément interpolation operator is well defined on L 1 (Ω). 2 If v P 1 (ω x ), then (π x )v(x) = v(x), x ω x. 3 If v P 1 ( ω ), then I h v(x) = v(x), x. 4 It is in the above sense that the Clément interpolation operator is polynomial (more precisely P 1 ) invariant. 9 / 37

10 The Residual and Error of Finite Element Solutions Error Estimates of the Clément Interpolation Operator I h Lemma There exist constants C 1 (θ min ) and C 2 (θ min ), which depend only on the smallest angle θ min of the triangular elements in the triangulation T h (Ω), such that, for any given T h (Ω), E E h and v V, v I h v 0,2, C 1 (θ min ) h v 1,2, ω, v I h v 0,E := v I h v 0,2,E C 2 (θ min ) h 1/2 v 1,2, ω E. 10 / 37

11 The Residual and Error of Finite Element Solutions Error Estimates of the Clément Interpolation Operator I h More general properties and proofs on the Clément interpolation operator may be found in [8, 31]. The basic ingredients of the proof are the scaling techniques (which include the polynomial quotient space and equivalent quotient norms, the relations of semi-norms on affine equivalent open sets), and the inverse inequality. 11 / 37

12 The Residual and Error of Finite Element Solutions An Upper Bound for the Dual Norm of the Residual R(u h ) Lemma There exists a constant C(θ min ), where θ min is the smallest angle of the triangular elements in the triangulation T h (Ω), such that f w dx + g w ds u h w dx Ω Ω 1 Ω { C(θ min ) w 1,2,Ω h 2 f 2 0,2, + + T h (Ω) E E h,ω h E [ν E u h ] E 2 0,E h E g ν E u h 2 0,E E E h,1 } 1/2, w V, 12 / 37

13 The Residual and Error of Finite Element Solutions Proof of the Lemma An Upper Bound for the Residual R(u h ) where in the theorem, ν E is an arbitrarily given unit normal of E if E E h,ω, and is the unit exterior normal of Ω if E E h,1, [ϕ] E is the jump of ϕ across E in the direction of ν E, i.e. Proof: [ϕ] E (x) = lim t 0+ ϕ(x + tν E ) lim t 0+ ϕ(x tν E ), x E. 1 Since u h is the finite element solution, we have R(u h )(v h ) := fv h dx+ gv h ds u h v h dx = 0, v h V h. Ω Ω 1 Ω In particular, R(u h )(w) = R(u h )(w I h w), for all w V. 13 / 37

14 The Residual and Error of Finite Element Solutions Proof of the Lemma An Upper Bound for the Residual R(u h ) 2 Applying the Green s formula on every element, denoting ν as the unit exterior normal of, and noticing u h P 1 () and thus u h = 0 on each element, we obtain u h v dx = u h v dx Ω = E E h,1 = T h (Ω) E { T h (Ω) u h v dx + ν E u h v ds+ E E h,ω E } ν u h v dx [ν E u h ] E v ds, v V. 14 / 37

15 The Residual and Error of Finite Element Solutions Proof of the Lemma An Upper Bound for the Residual R(u h ) Ω 3 Thus, recall R(u h )(w) = R(u h )(w I h w), we are lead to f w dx+ g w ds u h w dx = f (w I h w) dx Ω 1 Ω T h (Ω) + (g ν E u h )(w I h w) ds [ν E u h ] E (w I h w) ds, E E E h,1 E E E h,ω 15 / 37

16 The Residual and Error of Finite Element Solutions Proof of the Lemma An Upper Bound for the Residual R(u h ) 4 By the Cauchy-Schwarz inequality and Lemma 8.1, we have f (w I h w) dx f 0,2, w I h w 0,2, C 1 h f 0,2, w 1,2, ω, E E (g ν E u h )(w I h w)ds g ν E u h 0,E w I h w 0,E C 2 h 1/2 E g ν E u h 0,E w 1,2, ωe, [ν E u h ] E (w I h w) ds [ν E u h ] E 0,E w I h w 0,E C 2 h 1/2 E [ν E u h ] E 0,E w 1,2, ωe. 16 / 37

17 The Residual and Error of Finite Element Solutions Proof of the Lemma An Upper Bound for the Residual R(u h ) 5 The number of element in ω x, ω x C 3 = 2π/θ min, x N h. 6 Each element has three vertices, each edge E has two vertices. 7 Therefore, ω 3C 3, T h, ω E 2C 3, E E h. 8 Thus, we have [ w 2 1,2, ω + T h (Ω) E E h,ω E h,1 w 2 1,2, ω E ] 1/2 5 C3 w 1,2,Ω. 9 The conclusion of the lemma follows as a consequence of 3, 4 and 8 with C(θ min ) = 5C 3 max{c 1 (θ min ), C 2 (θ min )}. 17 / 37

18 A Theorem on the a Posteriori Error Estimate Theorem As a corollary of Theorem 8.1 and Lemma 8.2, we have the following a posteriori error estimate of the finite element solution: { u u h 1,2,Ω C h 2 f 2 0,2, + T h (Ω) E E h,1 h E g ν E u h 2 0,E + E E h,ω h E [ν E u h ] E 2 0,E } 1/2, where C = C(θ min ) C(Ω) is a constant depending only on Ω and the smallest angle θ min of the triangulation T h (Ω). The righthand side term above essentially gives an upper bound estimate for the V -norm of the residual R(u h ), which can be directly used as a posteriori error estimator for the upper bound of the error of the finite element solution.

19 A Residual Type A Posteriori Error Estimator A Practical a Posteriori Error Estimator 1 For convenience of analysis and practical computations, f and g are usually replaced by some approximation functions, say by f = 1 f dx and g E = h 1 E E g ds. 2 A practical a posteriori error estimator of residual type: η R, = {h 2 f 2 0,2, + h E g E ν E u h 2 0,E E E() E h,1 + 1 h E [ν E u h ] E 2 0,E 2 E E() E h,ω 3 In applications, f and g E can be further replaced by the numerical quadratures of the corresponding integrals. } 1/2. 19 / 37

20 Error Estimate Based on the Practical a Posteriori Error Estimator Theorem For the constant C = C(θ min ) C(Ω) in Theorem 8.2, the following a posteriori error estimate holds: { u u h 1,2,Ω C ηr, 2 + T h (Ω) T h (Ω) h 2 f f 2 0,2, + E E h,1 h E g g E 2 0,E } 1/2. Remark: Generally speaking, for h sufficiently small, the first term on the righthand side represents the leading part of the error. Therefore, in practical computations, η R, alone is often used to estimate the local error, particularly in a mesh adaptive process.

21 A Residual Type A Posteriori Error Estimator Reliability of an a Posteriori Error Estimator 1 The a posteriori error estimators given in Theorem 8.2 and 8.3 provide upper bounds for the error of the finite element solution u h in the V-norm. 2 Such a property is called the reliability of the a posteriori error estimator. 3 In general, the reliability of an a posteriori error estimator can be understood in the sense of a constant times 21 / 37

22 A Residual Type A Posteriori Error Estimator Reliability of an a Posteriori Error Estimator Definition Let u and u h be the solution and the finite element solution of the variational problem. Let η h be an a posteriori error estimator. If there exists a constant Ĉ independent of h such that u u h 1,2,Ω Ĉ η h. Then, the a posteriori error estimator η h is said to be reliable, or has reliability. 1 Reliability guarantees the accuracy. 2 To avoid mesh being unnecessarily refined and have the computational cost under control, efficiency is required. 22 / 37

23 A Residual Type A Posteriori Error Estimator Efficiency of an a Posteriori Error Estimator Definition Let u and u h be the solution and the finite element solution of the variational problem. Let η h be an a posteriori error estimator. If, for any given h 0 > 0, there exists a constant C(h 0 ) such that C(h 0 ) 1 u u h 1,2,Ω η h C(h 0 ) u u h 1,2,Ω, h (0, h 0 ), Then, the a posteriori error estimator η h is said to be efficient, or has efficiency. In addition, if the constant C(h 0 ) is such that lim C(h 0 ) = 1, h 0 0+ Then, the a posteriori error estimator η h is said to be asymptotically exact. 23 / 37

24 A Residual Type A Posteriori Error Estimator a Posteriori Local Error Estimator and the Local Error 1 In applications, to efficiently control the error, we hope to refine the mesh only on the regions where the local error is relatively large. 2 Therefore, in addition to a good estimate of the global error of a finite element solution, what we expect more on an a posteriori error estimator is that it can efficiently evaluate the local error. 24 / 37

25 A Residual Type A Posteriori Error Estimator a Posteriori Local Error Estimator and the Local Error 3 Recall Ω (u u h) w dx = R(u h )(w), and the a posteriori local error estimator of residual type is given as η R, = {h 2 f 2 0,2, + h E g E ν E u h 2 0,E E E() E h,1 + 1 h E [ν E u h ] E 2 0,E 2 E E() E h,ω } 1/2. 4 We hope, by choosing proper test functions w, to establish relationship between the local error of u u h and the three terms in η R,. 25 / 37

26 A Residual Type A Posteriori Error Estimator Relate Terms in η R, to the Local Errors of u u h 1 Notice that f is piecewise constant, w V, we have ( ) f (f w) dx = f 2 w dx = 1 w dx f 2 0,2,, 2 If we take a positive w H 1 0 (), called a bubble function on, then, the above equation will establish a relation between f 2 0,2, and the local error of (u u h) through (u u h ) (f w) dx = R(u h )(f w) = f (f w) dx = f (f w) dx + (f f ) (f w) dx. 26 / 37

27 A Residual Type A Posteriori Error Estimator Relate Terms in η R, to the Local Errors of u u h 3 Similarly, by taking proper bubble functions, we can also establish the relationship between the local error of u u h and the terms g E ν E u h 2 0,E, [ν E u h ] E 2 0,E, which are also piecewise constant functions. 27 / 37

28 A Residual Type A Posteriori Error Estimator Triangular Element Bubble Functions and Edge Bubble Functions 1 Let λ,i, i = 1, 2, 3 be the area coordinates of T h (Ω), define the triangular bubble function b as b (x) = { 27 λ,1 (x) λ,2 (x) λ,3 (x), x ; 0, x Ω \. 28 / 37

29 A Residual Type A Posteriori Error Estimator Triangular Element Bubble Functions and Edge Bubble Functions 2 For a given edge E E h,ω, let ω E = 1 2, let λ i,j, j = 1, 2, 3 be the area coordinates of i, denote the vertex of i which is not on E as the third vertex of i, define the edge bubble function b E as b E (x) = { 4 λ i,1(x) λ i,2(x), x i, i = 1, 2; 0, x Ω \ ω E. 29 / 37

30 A Residual Type A Posteriori Error Estimator Triangular Element Bubble Functions and Edge Bubble Functions 3 For a given edge E E h, Ω, let ω E =, denote the vertex of which is not on E as the third vertex of, define the edge bubble function b E as b E (x) = { 4 λ,1(x) λ,2(x), x ; 0, x Ω \. 30 / 37

31 A Residual Type A Posteriori Error Estimator Properties of the Bubble Functions Lemma For any given T h (Ω) and E E h, the bubble functions b and b E have the following properties: supp b, 0 b 1, max x b (x) = 1; supp b E ω E, 0 b E 1, max x E b E (x) = 1; E b E ds = 2 3 h E ; 31 / 37

32 A Residual Type A Posteriori Error Estimator Properties of the Bubble Functions Lemma there exists a constant ĉ i, i = 1,..., 6, which depends only on the smallest angle of the triangular triangulation T h (Ω), such that ĉ 1 h 2 b dx = 9 20 ĉ 2 h 2 ; ĉ 3 h 2 E b E dx = 1 3 ĉ 4 h 2 E, ω E ; b 0,2, ĉ 5 h 1 b 0,2, ; b E 0,2, ĉ 6 h 1 E b E 0,2,, ω E. 32 / 37

33 A Residual Type A Posteriori Error Estimator Proof of the Efficiency of η R, Estimate of f 0,2, 1 For any given T h (Ω), set w := f b. Then, by the properties of b (see Lemma 8.3), we have f w dx = 9 20 f 2 = 9 20 f 2 0,2,. 2 Since supp w, it follows g w ds u h w dx = u h Ω 1 Ω w dx = / 37

34 A Residual Type A Posteriori Error Estimator Proof of the Efficiency of η R, Estimate of f 0,2, 3 Thus, by Ω (u u h) w dx = R(u h )(w ), we obtain f w dx = = f w dx + (f f ) w dx (u u h ) w dx + (f f ) w dx u u h 1,2, w 0,2, + f f 0,2, w 0,2,. 34 / 37

35 A Residual Type A Posteriori Error Estimator Proof of the Efficiency of η R, Estimate of f 0,2, 4 On the other hand, since f is a constant, by the properties of b (see Lemma 8.3), we have ( 1/2 9 w 0,2, = f b 0,2, f b dx) = 20 f 0,2, ; w 0,2, ĉ 5 h 1 w 0,2,. 35 / 37

36 A Residual Type A Posteriori Error Estimator Proof of the Efficiency of η R, Estimate of f 0,2, 5 Combining the three inequalities obtained in 3 and 4 with f w dx = 9 20 f 2 0,2, (see (8.2.3)) leads to f 0,2, 9 ĉ5 h 1 u u h 1,2, + 9 f f 0,2,. Similar techniques can be applied to estimate the other terms in η R,. 36 / 37

37 S 8µ4. Thank You!