Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University
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
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
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.
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.
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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 = 0. 33 / 37
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
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
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 20 20 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
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