Validation of the a posteriori error estimator based on polynomial preserving recovery for linear elements

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2004; 61: ublished online 18 October 2004 in Wiley InterScience ( DOI: /nme.1134 Validation of the a posteriori error estimator based on polynomial preserving recovery for linear elements Zhimin Zhang,, and Ahmed Naga Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A. SUMMARY The purpose of this work is to investigate the quality of the a posteriori error estimator based on the polynomial preserving recovery (R). The main tool in this investigation is the computer-based theory. Also, a comparison is made between this estimator and the one based on the superconvergence patch recovery (SR). The results of this comparison were found to be in favour of the estimator based on the R. Copyright 2004 John Wiley & Sons, Ltd. KEY WORDS: finite element method; a posteriori error estimator; least-squares fitting; superconvergence; effectivity index; robustness index 1. INTRODUCTION A posteriori error estimation is crucial in adaptive finite element methods. Generally speaking, error estimators can be classified under two categories: the residual type (as in Reference [1]) and the recovery type (as in References [2, 3]). In the latter, a postprocessing technique is used to construct a recovered solution (as in Reference [4]) or a recovered gradient (as in References [5, 6]). Indeed, estimators based on gradient recovery are more effective than the ones based on solution recovery. For more comprehensive details about error estimation, the reader is referred to References [7 9] and the references therein. Among gradient recovery techniques, the superconvergence patch recovery (SR) [5] has been the most popular one for many years. Using the SR-recovered gradient in the Zienkiewicz Zhu (ZZ) error estimator [2] produces the robust estimator ZZ SR [10]. In the SR, the gradient of the finite element solution is postprocessed to construct the SR-recovered Correspondence to: Z. Zhang, Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A. zzhang@math.wayne.edu artially supported by the National Science Foundation Grants DMS , DMS and DMS anaga@math.wayne.edu Contract/grant sponsor: National Science Foundation; contract/grant numbers: DMS , DMS and DMS Received 12 May 2003 Revised 31 March 2004 Copyright 2004 John Wiley & Sons, Ltd. Accepted 30 April 2004

2 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1861 gradient. Naturally, one may wonder why not to postprocess the finite element solution itself. In an attempt to answer this question, the authors introduced the polynomial preserving recovery (R) in Reference [6] to recover the gradient from the finite element solution. The R has many good properties as was shown in References [6, 11]. For example it preserves the gradient of polynomials of certain degrees on any mesh. Also, the R-recovered gradient enjoys superconvergence in both structured and mildly unstructured meshes, and its stability is guaranteed under some mild geometric conditions. These results motivated the authors to use the R-recovered gradient in the ZZ error estimator. The resulting estimator (ZZ R) was tested on a set of benchmarks and was found to be as good as or better than the ZZ SR. However, benchmark computations may lead to inaccurate conclusions as was shown through examples in Reference [12]. The methodology introduced in Reference [13] provides a better way to validate an a posteriori error estimator. Using this methodology, many known estimators were studied in Reference [8, Section 5.7*] and [12] where it was found that the ZZ SR estimator is the most robust among the tested estimators. A summary of these tests and their results can be found in Reference [14]. The goal of this paper is to use the tests in Reference [7, Section 7.6] and [12] to validate the ZZ R through a comparison with the ZZ SR. As a side product, we theoretically verify some properties of the boundary layer that were observed in the numerical experiments conducted in Reference [12] (see Section 4). This work considers only the linear finite elements as the R is still in its early stages. In fact the R has more advantages over the SR both in higher order elements and in higher dimensional problems. This will be addressed in forthcoming papers Model problem The boundary value problem 2. RELIMINARIES Lu = (D u) = f in Ω ν (D u) = g on Γ N (1) u = 0 on Γ D models steady state heat conduction in an orthotropic medium Ω R 2 where Ω is a bounded polygonal domain. The boundary Ω = Γ N Γ D, ν is the unit outward normal vector to Ω, D is a constant symmetric positive definite matrix that represents the thermal conductivity, and the segments Γ N and Γ D are disjointed. Without loss of generality, assume that the principal thermal conductivities are 1 and d 1. If the orientation of the material orthotropy principal axes with respect to the problem co-ordinate system is θ, then D = ( d ) + ( d 1 2 ( d 1 2 ) sin(2θ) ) cos(2θ) ( ) d 1 sin(2θ) 2 ( ) ( ) d + 1 d 1 cos(2θ) 2 2 (2)

3 1862 Z. ZHANG AND A. NAGA If Γ N = Ω, the compatibility condition Ω f + Ω g = 0 must be satisfied and the condition Ω u = 0 is used to get a unique solution. As usual, H m (Ω) is the classical Sobolev space of order m equipped with the norm m,ω and the seminorm m,ω. The set of all polynomials defined on Ω R 2 of total degree r will be denoted by r (Ω ). In the variational form of (1), we seek u V ={v H 1 (Ω) : v = 0onΓ D } such that where B(u, v) = B(u, v) = l(v) v V (3) Ω D u v and l(v) = Ω fv+ gv Γ N Let T h be a triangulation of Ω and let N h denote the set of mesh nodes. In linear finite element methods, the finite element space S h associated with T h is defined by S h ={v C( Ω) : v 1 (τ) for every triangle τ T h } The finite element approximation u h S h of u is computed by restricting v in (3) to S h V, i.e. B(u h,v)= l(v) for all v S h V (4) For Ω Ω, we define the subspace S h (Ω ) and the bilinear operator B Ω, where S h (Ω ) ={v Ω : v S h } and B Ω (u, v) = D u v Ω 2.2. The definitions of the R and the ZZ R As we know, u h is discontinuous across elements boundaries. To better approximate u, gradient recovery techniques, like the SR and the R, constructs a continuous gradient in S h S h ; called the recovered gradient. Let G h : S h S h S h denote the R recovery operator. Since any function in S h is completely defined by its nodal values, G h u h takes the form, G h u h = (G h u h )(z) z z N h where { z : z N h } is the standard Lagrange basis of S h. Hence, it suffices to define G h u h at mesh nodes. The construction of G h u h proceeds in two stages. Stage 1 In this stage we consider internal mesh nodes. As in the SR, we use a patch ω z consisting of the triangles attached to z. Next, we find a polynomial p 2 (ω z ) that best fits, in least-squares sense, u h at the mesh nodes in ω z. Finally, G h u h (z) is defined by G h u h (z) = p(z) For uniqueness of p, ω z must have at least 6 nodes. If this is not the case and if ω z does not have any common edges with Ω, then ω z is extended out by adding every triangle sharing an edge with ω z as shown in Figure 1; otherwise recovering the gradient

4 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1863 Figure 1. Examples for patches used in the R recovery at some internal mesh nodes. The patch to the right has sufficient nodes in the first layer of elements while the one to the left does not. Figure 2. An example for an internal node that cannot be processed in Stage 1. at z is delayed to Stage 2. An example for nodes delayed to the second stage is shown in Figure 2. Stage 2 Basically this stage uses the gradient recovered in the first stage and linear extrapolation to complete the construction of G h u h in a finite number of iterations. A typical iteration starts by defining T h,0 T h where a mesh triangle τ T h,0 if and only if G h u h is defined at all the vertices of τ. For an unprocessed node z, let ω z be the patch consisting of the triangles attached to z. We have two cases. 1. The patch ω z has common edges with τ 1, τ 2,...,τ Nz T h,0. Define the patch ω τi = τ i {τ T h,0 : τ τ i is an edge} for i = 1, 2,...,N z Note that G h u h is well defined on ω τi and that ω τi has at least 4 nodes. Using discrete least-squares, we can find the polynomial q x,i 1 (ω τi ) that best fits the x components of G h u h at the nodes in ω τi. The recovered x derivative at z is defined Nz to be 1/N z i=1 q x,i(z). Similarly, one can define the recovered y derivative at z. 2. The patch ω z has no common edges with any triangle in T h,0. In this case z is left for the next iteration. At the end of the iteration, if all the nodes have been processed, we are done; otherwise we start another iteration.

5 1864 Z. ZHANG AND A. NAGA Figure 3. Recovering the gradient at boundary nodes in the R technique. Figure 3(a) shows some examples that explain the order in which the mesh nodes are processed in the R. Nodes labelled with 0 are those processed in the first stage. Nodes processed in the second stage are labelled 1, 2, 3, or 4, depending on the iteration in which they are processed. Let us have an example that will explain the computations in the second stage. The example is depicted in Figure 3(b) where we want to define G h u h at z Ω. The triangles attached to z are τ 1, τ 2, and τ 3. Both τ 1 and τ 3 have no common edges with triangles in T h,0,butτ 2 shares a common edge with τ 4 T h,0.ifweuseτ 4 by itself in extrapolating the gradient at z, the extrapolation may be unstable especially in severely distorted meshes. Therefore, we use the triangles τ 5 and τ 6 along with τ 4 to carry out the extrapolation. The triangles τ 5 and τ 6 are chosen because they are in T h,0 and every one of them shares an edge with τ 4. Next, we

6 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1865 compute two linear polynomials that best fit, in least-squares sense, the components of G h u h at the nodes in ω τ4 = τ 4 τ 5 τ 6. Finally, the computed polynomials are evaluated at z to get G h u h (z). Remark For a comprehensive discussion about the properties of G h, the reader is referred to References [6, 11]. The most important property of G h is that it preserves quadratic polynomials. In other words, if Ω Ω, then G h (p I ) = p p 2 (Ω ) where p I S h (Ω ) is the Lagrange interpolation of p. Note that this property is valid up to the boundary. 2. For the definitions of the SR-recovered gradient at mesh nodes, the reader is referred to Reference [5] and [12, A.3.1]. In this work, we will follow the definitions used in the latter reference. We will adopt the following notations from the literature, see e.g. Reference [8, Section 5.7]. Let η(u, u h, Ω ) denote the error estimated with the ZZ R over Ω Ω, i.e. η(u, u h, Ω ) = G h u h u h L2 (Ω ) The reliability of η(u, u h, Ω ) is measured using the effectivity index κ(u, u h, Ω ) where κ(u, u h, Ω η(u, u h, Ω ) ) = (u u h ) L2 (Ω ) By varying u over a class of solutions and varying T h over a collection of meshes, it is more realistic to have κ(ω ) lim κ(u, u h, Ω ) κ(ω ) h 0 These two asymptotic bounds measure the robustness of the ZZ R through the robustness index R where Clearly, R(Ω ) 0 where 0 is the ideal value. { } R(Ω ) = max κ(ω 1 ) 1, κ(ω ) 1 3. ASYMTOTIC ERFORMANCE OF THE ZZ R INSIDE Ω In this section we evaluate the asymptotic performance of the ZZ R in internal patches. Also, we compare the ZZ R with the ZZ SR. The evaluation is based on a collection of tests that were used in Reference [7, Section 7.6] and [12]. In fact, the tests are products of the computer-based theory [15] and a methodology to study the asymptotic performance of a posteriori error estimators developed by Babuška et al. [13]. According to this methodology the asymptotic value of the robustness of the ZZ R in linear elements is computed by considering homogeneous quadratic polynomials on a translationinvariant domain tiled with scaled translations of a reference cell C. Moreover, this asymptotic

7 1866 Z. ZHANG AND A. NAGA value is the same on any cell in this domain. Like many error estimators, including the ZZ SR, the ZZ R is scale invariant. As a result, the computations can be done in C after embedding it in a reference domain ˆΩ as shown in Figure 4(a). For the theoretical details, the reader is referred to Reference [13, pp ], [7, pp ], and [16, Chapter 12]. Before we continue, we need the following notations, definitions, and a general setup from the literature, see e.g. Reference [8, pp ] and [7, pp ]. Define the finite element spaces S(C) ={ˆv H 1 (C) :ˆv τ 1 (τ) τ T C }, S per (C) = S(C) H 1,per (C) where T C is the triangulation of C and H 1,per (C) H 1 (C) is the subspace of functions satisfying the periodicity conditions (a function in H 1 (C) satisfies the periodicity conditions if it the same on opposite sides of C). The projection of û H 1,per (C) onto S per (C) is ŵ = Π per C û such that B C (û ŵ, ˆv) = 0 (û ŵ) = 0 C ˆv S per (C) Consider a quadratic polynomial ˆq = 3 j=1 c j ˆq j where ˆq 1 = ˆx 2, ˆq 2 = ˆxŷ, ˆq 3 = ŷ 2, and c 1,c 2,c 3 are real constants. Let ˆq I be the Lagrange interpolation of ˆq on ˆΩ and define the function ˆq =ˆq I + E per ˆΩ ˆqper (5) where ˆq per = Π per C ( ˆq ˆq I ) and E per ˆΩ periodically extends functions in H 1,per (C) to all of ˆΩ. Let G ˆq denote the R-recovered gradient of ˆq. Since ˆq = 3 j=1 c j ˆq j and G ˆq = 3 j=1 c j G ˆq j, it is straightforward to verify that G ˆq ˆq 2 L 2 (C) = ct M e c, ( ˆq ˆq ) 2 L 2 (C) = ct M a c where c =[c 1 c 2 c 3 ] T, and M e, M a R 3 3. Consequently, κ( ˆq, ˆq, C) = F(c) where F : R 3 R is defined by F(c) = ct M e c c T M a c Indeed, and as was shown in Reference [13], (6) lim κ(u, u h, Ω ) = κ( ˆq, ˆq, C) h 0 where the homogeneous quadratic part of the Taylor expansion of u at the centroid of Ω is a scaled translation of ˆq. This result is true if u and u u h satisfy certain conditions, Ω is inside Ω Ω, and T h in Ω is the union of h-scaled translations of T C. Since M e and M a are symmetric positive definite matrices, λ min F(c) λ max

8 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1867 ŷ ˆ Ω zˆ = ( 1, a) 7 zˆ = (0, a) 8 zˆ = (1, a) 9 ˆx z ˆ 4 = ( 1,0) z ˆ 5 = (0,0) z ˆ 6 = (1, 0) zˆ = ( 1, a) 1 zˆ = (0, a) 2 zˆ = (1, a) 3 (a) (b) Figure 4. ˆΩ, C, and T C for Example 1: (a) the reference domain ˆΩ; and (b) mesh nodes and triangles in C. where λ min and λ max are, respectively, the minimum and the maximum eigenvalues of the generalized eigenvalue problem M e c = λm a c (7) Based on this general setup, some examples are explicitly constructed in Reference [12]. Following this setup, the next example is particularly constructed for ZZ R. Being needed in Section 4, the computations are given in detail. Example 1 Consider the reference cell C =[ 1, 1] [ a,a] and a is the aspect ratio of C. The mesh T C partitions C into triangles arranged in the Chevron pattern as shown in Figure 4(b). Let us see how to compute qˆ 1 when D = I 2, the identity of R 2 2. To simplify notations, let û =ˆx 2 and let û I be its Lagrange interpolation over ˆΩ. Then, û =û I + E per ˆΩ ŵ (8) where ŵ = Π per C ê and ê = (û û I ) C. Then, B C (ŵ, ˆv) = B C (ê, ˆv) ˆv S per (C) (9) ŵ = ê C C Set û i =û(ẑ i ), set ŵ i = ŵ(ẑ i ), and let ˆφ i be the standard Lagrange basis function associated with the ith node where i = 1, 2,...,9. Then, û I C = 9 û i ˆφ i = ˆφ T û L and ŵ = 9 ŵ i ˆφ i = ˆφ T ŵ L (10) i=1 i=1

9 1868 Z. ZHANG AND A. NAGA where û L =[û 1 û 2 û 9 ] T, ŵ L =[ŵ 1 ŵ 2 ŵ 9 ] T, and ˆφ =[ˆφ 1 ˆφ 2 ˆφ 9 ] T. Applying the standard finite element sub-assembling procedure on the first equation in (9) leads to K L ŵ L = ê L (11) where a 2 +1 a a 2 2(a 2 +1) a a 2 a (a 2 +1) 2a K L = a 2a 2 4(a 2 +1) 2a a 2 2(a 2 +1) a 2 +1 a a 2 2(a 2 +1) a a 2 a 2 +1 and ê L = a [ ]T 6 Since ŵ S per (C), ŵ satisfies the periodicity conditions. Therefore, ŵ 3 =ŵ 7 = ŵ 9 = ŵ 1 ŵ 8 =ŵ 2 (12) ŵ 6 =ŵ 4 Let ŵ =[ŵ 1 ŵ 2 ŵ 4 ŵ 5 ] T, then (12) implies that ŵ L = R T ŵ where R = re-multiplying the linear system in (11) by R, weget K ŵ = 0 (13)

10 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1869 where a a K = RK L R T = 2 a 2 a a 1 0 a a a 2 a Note that K is singular. Using (10) in the second condition of (9) leads to ( ) ( ) φ T ŵ L = û φ T û L a 6 [ ]ŵ L = 2a 3 C Using the relation between ŵ L and ŵ,weget C a[1 111]ŵ = 2a 3 Solving equations (13) and (14) ŵ = 1 6 [1111]T. Hence, Similarly, and ˆq per C 1 = Π per C ( qˆ 1 ˆq 1,I ) = ŵ = 1 [ ] ˆφ 6 ˆq per 2 = Π per ( ˆq 2 ˆq 2,I ) =[ ] ˆφ C (14) ˆq per 3 = Π per C ( ˆq 3 ˆq 3,I ) = a2 [ ] ˆφ 6 These results imply that E per ˆΩ ˆqper is identically constant over ˆΩ for any homogeneous quadratic polynomial ˆq. Since G preserves the polynomials in 2 ( ˆΩ), weget G ˆq = G ˆq I + G(Π per ˆΩ ˆqper ) = ˆq + 0 = ˆq Therefore, M a = M e. Indeed, M a = M e = 2a a a 2 Thus, and unlike the ZZ SR, the ZZ R is asymptotically exact in case of the Chevron pattern for any aspect ratio a. We are now ready to evaluate the ZZ R. Let C =[ 1, 1] [ a, a] for some a>0. The mesh T C partitions C into triangles arranged into one of the well-known patterns shown in

11 1870 Z. ZHANG AND A. NAGA (a) (b) (c) (d) Figure 5. The four patterns used in partitioning C: (a) Regular; (b) Chevron; (c) Union Jack; and (d) Criss Cross. Figure 5. The ZZ R is evaluated through a collection of tests classified into the following three categories. 1. In the first category, we will study the effect of distorting T C when a = 1 and D = I 2. The mesh T C is distorted by moving the central node along one of the four lines ŷ = 0, ˆx = 0, ŷ =ˆx, orŷ = ˆx, as shown in Figure In the second category, we study the effect of varying the aspect ratio a when D = I In the third category, we study the effect of changing the material properties, represented in (2) by θ and d, when a = 1. In any of these tests, c will vary over R 3. This implies that the minimum and the maximum values of the effectivity index are λ min and λ max, respectively. There is a special situation that is worth considering. When D = I 2 and f = 0 in (1), i.e. u is harmonic. For the class of harmonic functions, the tests must be done for harmonic homogenous quadratic polynomials, i.e. when c 3 = c 1. In this special case, M e and M a are replaced with H T M e H and H T M a H, respectively, where 1 0 H =

12 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1871 (a) (b) (δ δ) (c) (d) Figure 6. Distorting T C from the Criss Cross pattern by moving the central node: (a) moving the central node along the line ŷ = 0; (b) moving the central node along the line ˆx = 0; (c) moving the central node along the line ŷ = ˆx; and (d) moving the central node along the line ŷ = ˆx. Remark 3.1 The distortion tests in the first category were motivated by the example given in Reference [7, Section 7.6]. The tests in the second and the third categories are some of the tests used in Reference [12] to study the performance of the a posteriori error estimators both inside Ω and near Ω. The response of the robustness index in the distortion tests is reported in Figures For better understanding, here we explain one of the distortion tests. If the Chevron pattern is distorted by moving the middle node along ŷ =ˆx to (δ, δ), then the topology of T C will not change as long as δ ( 1, 0.5). Of course, the admissible range of δ varies from one mesh pattern to another. Using a symbolic computing package, λ min and λ max, as well as the robustness index, can be determined in terms of δ. From the distortion tests, we can conclude the following remarks: The ZZ R is exact for δ = 0 in all of the four patterns even in the Chevron pattern where the ZZ SR fails to be exact. The ZZ R is less sensitive to δ than the ZZ SR especially when δ is small. Both the ZZ R and the ZZ SR maintain their robustness when some of the triangles in T C degenerate as δ reaches its limits.

13 1872 Z. ZHANG AND A. NAGA ŷ ˆΩ zˆ (2 m,1) z ˆ(2 m,2) zˆ (2 m,3) zˆ (2 m 1,1) zˆ (2 m 1,2) zˆ (2 m 1,3) (a) ˆ Γ ˆx (b) zˆ (2 m 2,3) zˆ (2 m 2,2) zˆ (2 m 2,3) Figure 7. ˆΩ, S, T S, and the mth cell for Example 2: (a) reference domain ˆΩ; and (b) mesh nodes in the mth cell. Figure 8. The change in the robustness index as T C is distorted from the Regular pattern internal patches.

14 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1873 Figure 9. The change in the robustness index as T C is distorted from the Regular pattern internal patches harmonic case. Figure 10. The change in the robustness index as T C is distorted from the Chevron pattern internal patches.

15 1874 Z. ZHANG AND A. NAGA Figure 11. The change in the robustness index as T C is distorted from the Chevron pattern internal patches harmonic case. Figure 12. The change in the robustness index as T C is distorted from the Union Jack pattern internal patches.

16 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1875 Figure 13. The change in the robustness index as T C is distorted from the Union Jack pattern internal patches harmonic case. Figure 14. The change in the robustness index as T C is distorted from the Criss Cross pattern internal patches.

17 1876 Z. ZHANG AND A. NAGA Figure 15. The change in the robustness index as T C is distorted from the Criss Cross pattern internal patches harmonic case. Table I. Response of λ min and λ max to changes in a,θ, and d internal patches. Changing factor Cell aspect ratio Material properties General polynomials Harmonic polynomials attern R SR R SR R SR Regular λ min = 1 λ min = 1 λ min = 1 λ min = 1 λ min = 1 λ min = 1 λ max = 1 λ max = 1 λ max = 1 λ max = 1 λ max = 1 λ max = 1 Chevron λ min = 1 λ min = 37a (a 2 +1) λ min = 1 λ min = 37a (a 2 +1) λ max = 1 λ max = λ max = 1 λ max = 50a (a 2 +1) λ min = 1 λ min = λ max = 1 λ max = Union Jack λ min = 1 λ min = 1 λ min = 1 λ min = 1 λ min = 1 λ min = 1 λ max = 1 λ max = 1 λ max = 1 λ max = 1 λ max = 1 λ max = 1 Criss Cross λ min = 1 λ min = 1 λ min = 1 λ min = 1 λ min = 1 λ min = 1 λ max = 1 λ max = 1 λ max = 1 λ max = 1 λ max = 1 λ max = 1 The robustness of the ZZ SR improves when the tests target the class of harmonic polynomials. The same is true for the ZZ R although the change is not significant in the case of the Regular pattern.

18 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1877 Table I reports the response of the effectivity index to changes in aspect ratio a and to changes in material properties θ and d. It is clear from the table that the ZZ R is asymptotically exact in all of the four patterns regardless of a,θ, and d even in the Chevron pattern in which the ZZ SR fails to be exact. 4. THE BOUNDARY LAYER In general, the approximation ˆq, as defined in (5), does not satisfy the boundary conditions. Hence, ˆq cannot be used in validating an a posteriori error estimator over patches adjacent to Ω. To overcome this problem, Babuška et al. [12] modified the definition of ˆq by adding a new component, called the boundary layer, such that the modified ˆq satisfies the boundary conditions. Moreover, this boundary layer must decay as one gets far from the boundary in order to maintain the validity of the scheme used in studying error estimators inside the domain. As in Section 3, it suffices to do the computations in a reference domain ˆΩ. However, this time ˆΩ is infinite, as depicted in Figure 7(a), to take the decay of the boundary layer into account. Note that ˆΩ is the union of non-overlapping infinite vertical strips and each of them is a horizontal translation of the strip S =[ 1, 1] [0, ], and S is the union of vertical translations of a reference cell C. The cells in S are numbered as 1, 2,..., starting at the bottom and C 1 = C. The lower horizontal edge of C, denoted by ˆΓ, is along the ˆx-axis. Define the finite element spaces and S bl (S) ={ˆv :ˆv τ 1 (τ) τ T S, ˆv( 1, ŷ) =ˆv(1, ŷ) ŷ [0, )} S bl,0 (S) ={ˆv S bl (S) :ˆv ˆΓ = 0} where T S denotes the mesh in S. Note that a function ˆv S bl (S) can be extended to a continuous function in ˆΩ that is periodic in the horizontal direction. The extended function corresponding to ˆv will be denoted by E bl ˆv where Ebl is the extension operator. ˆΩ ˆΩ The modified definition of ˆq, after including the boundary layer, is ˆq =ˆq I + E per ˆΩ ˆqper + E bl ˆΩ ˆqbl (15) where ˆq bl S bl (S) is the boundary layer that satisfies the decay condition limŷ ˆq bl = 0. In Dirichlet boundary conditions, ˆq bl satisfies, B S ( ˆq bl, ˆv) = B S ( ˆψ, ˆv) ˆv S bl,0 (S) ˆq bl ˆΓ = ˆqper ˆΓ where ˆψ =ˆq ˆq I E per ˆΩ ˆqper, and in Neumann Boundary conditions ˆq bl satisfies B S ( ˆq bl, ˆv) = B S ( ˆψ, ˆv) ˆv S bl,0 (S) For the details about the previous setup, the reader is referred to Reference [12].

19 1878 Z. ZHANG AND A. NAGA roposition 4.1 B S ( ˆψ, ˆv) = 0 for all ˆv S bl,0 (S). roof Let ˆv S bl,0 (S) and assume its support is the union of the first k cells. Note that ˆv = 0on ˆΓ and on the upper edge of the kth cell. Since ˆψ S per (S), weget B S ( ˆψ, ˆv) = k m=1 ( B Cm ( ˆψ, ˆv) = B C1 ˆψ, k m=1 ) ˆv(ˆx,ŷ (m 1)l) where l is the height of C. It is easy to verify that k m=1 ˆv(ˆx,ŷ (m 1)l) S per (C 1 ). Hence, ( B S ( ˆψ, ˆv) = B C1 ˆψ, k m=1 ) ˆv(ˆx,ŷ (m 1)l) = 0 Since any ˆv S bl,0 (S) is a linear combination of functions in S bl,0 (S) with each of them having a support over a finite number of cells, the conclusion is true for all ˆv S bl,0 (S). Using roposition 4.1, and as was shown in Reference [12], ˆq bl is constructed as follows: 1. In case of Dirichlet boundary conditions, ˆq bl solves the problem, B S ( ˆq bl, ˆv) = 0 ˆq bl ˆΓ = ˆqper ˆΓ ˆv S bl,0 (S) (16) In case of Neumann boundary conditions, ˆq bl solves the problem, B S ( ˆq bl, ˆv) = B S ( ˆψ, ˆv) lim ŷ ˆqbl = 0 ˆv S bl (S) (17) The second condition in (17) is for uniqueness. 2. Regardless of the boundary conditions, ˆq bl satisfies the decay condition limŷ ˆq bl = 0. The following example illustrates how to compute ˆq bl and provides the outline of the proof of Theorem 4.5. For more general cases, the reader is referred to Reference [12, A.2]. Example 2 In this example we will compute ˆq 1 bl when C =[ 1, 1] [0, 2a] is triangulated using the Chevron pattern. The mesh nodes in the mth cell are numbered as shown in Figure 7(b).

20 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1879 First, consider Dirichlet boundary conditions case. To simplify notations, set ŵ =ˆq bl 1. Hence, B S (ŵ, ˆv) = 0 ˆv S bl,0 (S) ŵ ˆΓ = ˆqper 1 ˆΓ lim ŵ = 0 ŷ (18) For j 0, set ŵ (j,i) = ŵ(ẑ (j,i) ) for i = 1, 2, 3, and consider the vectors ŵ (j) =[ŵ (j,1) ŵ (j,2) ] T and ŵ (j) L =[ŵ (j,1) ŵ (j,2) ŵ (j,3) ] T Using Example 1 and the boundary conditions in (18) ŵ (0) L = 1 6 [111]T. The stiffness matrix and the load vector resulting from the finite element sub-assembling in the mth cell are K L and 0, respectively, where K L is as in Example 1. Since ŵ S bl (S), ŵ (j,3) = ŵ (j,1) j 0. Hence, ŵ (2m 2) L ŵ (2m 2) ŵ (2m 1) L = RT ŵ (2m 1) (19) ŵ (2m) L ŵ (2m) where R = Using (19) the stiffness matrix K L is reduced to K where a a a 2 a K = RK L R T = (a 2 + 1) 2a a 0 1 2a 2 2(a 2 + 1) a a a 2 a 2 + 1

21 1880 Z. ZHANG AND A. NAGA Note that the degrees of freedom ŵ (2m 1) appears only in the mth cell and the load vector is zero. Hence, using the third and fourth rows of K, it can be shown that [ ŵ (2m 1) 1 a a 2 ] = 2(2a 2 (ŵ (2m 2) + 1) a 2 a 2 + ŵ (2m) ) (20) + 1 Using (20), the degrees of freedom ŵ (2m 1) are eliminated and K is reduced to K where [ ] K1 K K = 2 K 2 K 1 = 1 2a(2a 2 + 1) 4a 4 + 5a a 2 (4a 2 + 3) (a 2 + 1) a 2 a 2 (4a 2 + 3) 4a 4 + 5a a 2 (a 2 + 1) (a 2 + 1) a 2 4a 4 + 5a a 2 (4a 2 + 3) a 2 (a 2 + 1) a 2 (4a 2 + 3) 4a 4 + 5a and K 1,K 2 R 2 2. Assembling the matrices K from mth and (m + 1)th cells, we get K 2 ŵ (2m 2) + 2K 1 ŵ (2m) + K 2 ŵ (2m+2) = 0 for m 1 (21) ŵ (0) = 1 6 [11]T (22) The solution of the recurrence relation in (22) when m 2 can be written in the form, ŵ (2j) = μ j b for j 1 (23) where μ R and b R 2. To determine μ and b, use (23) in the recurrence equation in (22). After simplification, we get 2K 1 b = K 2 (μ 1 + μ)b (24) Setting A = K2 1 K 1, (24) takes the form Ab = ((μ 1 + μ)/2)b. Hence, b is an eigenvector of A and its corresponding eigenvalue is (μ 1 + μ)/2. The eigenvalues of A are 1 and (1 + 8a 2 + 8a 4 ), and their corresponding eigenvectors are b 1 =[11] T and b 2 =[ 11] T, respectively. The decay condition imposed on ŵ implies μ ( 1, 1]. Therefore, the possible values of μ are μ 1 = 1 and μ 2 = (1 + 8a 2 + 8a 4 ) 4a(1 + 2a 2 ) 1 + a 2, and ŵ (2j) = γ 1 μ j 1 b 1 + γ 2 μ j 2 b 2 for j 1 for some constants γ 1, γ 2 R. Note that μ 2 = μ 2 (a) (0, 1] for all a [0, ) as 1 + 8a 2 + 8a 4 4a(1 + 2a 2 ) = 1 + 2a2 4a + a(1 + a2 ) 1 + 2a 2 > 1 + a 2, 1 μ 2 (a) = 4a 1 + a 2 (a a 2 ) 2

22 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1881 Using (22) when m = 1 leads to [ ] K 2 ŵ (0) = 2K 1 ŵ (2) + K 2ŵ (4) = 2 γ i [2μ i K 1 + μ 2 i K 2]b i i=1 By (24), and by invertibility of K 2, the last equation is simplified to the form, [ ] = 2 γ1 γ i b i =[b 1 b 2 ] ŵ (0) Solving this linear system, we get γ 1 = 1 6, γ 2 = 0, and ˆq bl = 1 6 everywhere on S. Next, we consider the Neumann Boundary conditions case. In this case i=1 γ 2 B S (ŵ, ˆv) = B S ( ˆψ 1, ˆv) lim ŵ = 0 ŷ lim ŵ = 0 ŷ ˆv S bl (S) (25) where ˆψ 1 =ˆq 1 ˆq 1,I E per ˆΩ case if we assume that ŵ (0) ˆqper 1. We can proceed exactly as in Dirichlet boundary conditions is available. Hence, and after using the second condition in (25), ŵ (2j) = γ 2 μ j 2 b 2 for j 0 (26) where γ 2 R. Let ˆ i be the standard Lagrange basis function at z (0,i) for i = 1, 2, 3 (the nodes on ˆΓ). Note that both ( ˆ 1 + ˆ 3 ) and ˆ 2 are in S bl (S). If ˆv ˆΓ = 0, then ˆv = 1+ ˆ 1 + ˆ 2 + ˆ 3 +ĥ for some function ĥ S bl,0 (S). Since B S ( ˆψ 1, ˆv) = B S ( ˆψ 1, ĥ) = 0 by roposition 4.1, we get B S ( ˆψ 1, ˆv) = 0 B C1 ( ˆψ 1, ˆ 1 + ˆ 3 ) = B C1 ( ˆψ 1, ˆ 2 ) Using ( ˆ 1 + ˆ 3 ) and ˆ 2 in the first equation of (25), we get, K 1 ŵ (0) + K 2ŵ (2) = γ 2(K 1 + μ 2 K 2 )b 2 = B C1 ( ˆψ 1, ˆ 2 )b 2 (27) From (24), (K 1 + μ 2 K 2 )b 2 = (μ 2 μ 1 2 )K 2b 2 /2. Hence, (μ γ 2 μ 1 2 ) 2 K 2 b 2 = B C1 ( 2 ˆψ 1, ˆ 2 ) γ 2 = B C 1 ( ˆψ 1, ˆ 2 ) 2 a = 1 + a a 2 Remark 4.2 The result in Example 2 for the Dirichlet Boundary conditions case is expected. In general, if ˆq per ˆΓ = γ for some constant γ R, then ˆqbl = γ in S( ˆΩ). Note that this is the typical situation when T C has 2 mesh nodes on ˆΓ as in the Criss Cross pattern. Remark 4.3 Equation (26) sheds some light over the behaviour of the boundary layer in Example 2 under the Neumann boundary condition. The trivial case happens when a 0 as γ 2 0. As a

23 1882 Z. ZHANG AND A. NAGA moves from 0 to, μ 2 (a) monotonically decreases from 1 to 0 where lim a 16a 4 μ 2 (a) = 1. Indeed, μ 2 (1) = ( ) 1. Hence, the boundary layer term decays exponentially as we move away from the boundary. Indeed, these results agree with the numerical experiments conducted in Reference [12] which showed that the boundary layer has very little effect on the performance of a posteriori error estimators. Remark 4.4 In this work, C =[ 1, 1] [0, 2a] for some a>0 and T C is a distorted version of one of the patterns in Figure 5. The distortion is achieved by moving the central node in C without changing the topology of T C. In this setup, ˆq bl can be determined analytically. Next, we will develop a theory that explains the structure of the boundary layer component when C and T C are the same as in Remark 4.4. The proof of this theory is constructive and generalizes the results established in Example 2. For the moment, let us consider the Dirichlet boundary condition. In this case, the Criss Cross pattern, even after distortion, is trivial according to Remark 4.2. Therefore, we will focus on the other three patterns. As in Example 2, we set ŵ =ˆq bl and number the nodes in the mth cell as in Figure 7(b). Then, ŵ is the solution of (18) after replacing ˆq per 1 with ˆq per. Using the notations of Example 2 and following the steps carried out in the mth cell, we can compute K L, reduce K L to K, then reduce K to K. Since K is symmetric, [ K1 K T ] 2 K = (28) K 2 K 3 for some K 1,K 2,K 3 R 2 2. Assembling the mth and (m + 1)th cells, we get, K 2 ŵ (2m 2) + (K 1 + K 3 )ŵ (2m) + K T 2 ŵ(2m+2) = 0 for m 1 ŵ (0) = [ˆqper (ẑ (0,1) ) ˆq per (ẑ (0,2) )] T (29) Let E denote any horizontal edge in any cell in S. From the properties of ŵ, ŵ E is an even function of ˆx. Since any piecewise linear even function of ˆx [ 1, 1] is a linear combination of 1 and ( ˆx ), ŵ has two components. The restriction of the first component to E is a multiple of 1 while the restriction of the second component to E is a multiple of ( ˆx ). Consequently, ŵ (2j) = μ (1,j) b 1 + μ (2,j) b 2 for j 0 (30) where b 1 and b 2 are as in Example 2. The values μ (1,0) and μ (2,0) are determined from ŵ (0). By Remark 4.2 we can assume that μ (2,0) = 0. Since the components of ŵ are independent, each component in ŵ (2j) satisfies the recurrence equation in (29) for m 2. Consequently, μ (i,j 1) K 2 b i + μ (i,j) (K 1 + K 3 )b i + μ (i,j+1) K T 2 b i = 0 for j 2 and i = 1, 2 (31) re-multiplying (31) by b T i, we get, (μ (i,j 1) + μ (i,j+1) )b T i K 2b i + μ (i,j) b T i (K 1 + K 3 )b i = 0 for j 2 and i = 1, 2 (32)

24 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1883 Since (K 1 + K 3 ) is positive definite, as B S is coercive, then b T i (K 1 + K 3 )b i > 0 for i = 1, 2. The first equation in (18) is valid for any constant. Hence, K 2 b 1 + (K 1 + K 3 )b 1 + K T 2 b 1 = 0 b T 1 (K 1 + K 3 )b 1 = 2b T 1 K 2b 1 (33) Therefore, (32) implies (μ (1,j 1) + μ (1,j+1) ) 2μ (1,j) = 0 when i = 1. The general solution of this difference equation is μ (1,j) = γ 1 + γ 2 j for some γ 1, γ 2 R, and for j 1. Using the decay condition imposed on ŵ, γ 2 = 0. Next, consider (32) when i = 2. If b T 2 K 2b 2 = 0, then K 2 b 2 = γ 3 b 1 for some γ 3 R and 0 for j = 1 and γ 3 = 0 μ (2,j) = γ 2 for j = 1 and γ 3 = 0 (34) 0 forj >1 for some γ 2 R. Ifb T 2 K 2b 2 = 0, (32) takes the form (μ (2,j 1) +μ (2,j+1) ) 2ρμ (2,j) = 0 where 2ρ = b T 2 (K 1 + K 3 )b 2 /b T 2 K 2b 2. The characteristics of this difference equation are reciprocal of each other and their magnitudes are 1 if they are complex. By virtue of the decay condition, { 0 ρ 1,j 1 μ (2,j) = γ 2 [sign(ρ)( ρ ρ 2 1)] j (35) ρ > 1,j 1 for some γ 2 R. To determine γ 1 and γ 2, use (29) when m = 1. By virtue of (33), we get, (μ (1,0) γ 1 )K 2 b 1 + μ (2,0) K 2 b 2 + μ (2,1) (K 1 + K 3 )b 2 + μ (2,2) K T 2 b 2 = 0 (36) At this point, we have four cases. Case 1: Ifb T 2 K 2b 2 = 0 with K 2 b 2 = γ 3 b 1 for some γ 3 > 0, then (34) and (36) lead to (μ (1,0) γ 1 )K 2 b 1 + μ (2,0) γ 3 b 1 = 0 Hence, b 1 is an eigenvector of K 2 as μ (2,0) = 0 by assumption. re-multiplying by b T 1,we get, γ 1 = μ (1,0) + 2μ (2,0)γ 3 b T 1 K 2b 1 Case 2: Ifb T 2 K 2b 2 = 0 with K 2 b 2 = 0, then (34) and (36) lead to (μ (1,0) γ 1 )K 2 b 1 + γ 2 (K 1 + K 3 )b 2 = 0 and we have two unknowns. In this case, K 2 b 1 and (K 1 + K 3 )b 2 are independent as unique. Therefore, γ 2 = 0 and γ 1 = μ (1,0). Case 3: Ifb T 2 K 2b 2 = 0 with ρ 1, then (35) and (36) lead to, (μ (1,0) γ 1 )K 2 b 1 + μ (2,0) K 2 b 2 = 0 Since μ (2,0) = 0, we must have K 2 b 2 = γ 3 K 2 b 1 for some γ 3 R. Hence, ŵ is γ 1 = μ (1,0) + μ (2,0) γ 3

25 1884 Z. ZHANG AND A. NAGA Case 4: Ifb T 2 K 2b 2 = 0 with ρ > 1, then (31), when i = j = 2, and (35) imply Hence, (36) takes the form, γ 2 K 2 b 2 + μ (2,1) (K 1 + K 3 )b 2 + μ (2,2) K T 2 b 2 = 0 (μ (1,0) γ 1 )K 2 b 1 + (μ (2,0) γ 2 )K 2 b 2 = 0 By uniqueness of ŵ, K 2 must be non-singular. Consequently, γ i = μ (i,0) for i = 1, 2 The previous results are summarized in the following theorem. Theorem 4.5 Let C and T C be as in Remark 4.4 after excluding the cases in which T C is a distorted version of the Criss Cross pattern. Let ŵ be the boundary layer corresponding to ˆq in S assuming Dirichlet boundary conditions. Let b 1, b 2, and ŵ (2j) for j 0 be defined as in Example 2, and let K 1,K 2 and K 3 be defined as in (28). If b T 2 K 2b 2 = 0, set 2ρ = b T 2 (K 1 + K 3 )b 2 /b T 2 K 2b 2. Then, ŵ (2j) = μ (1,j) b 1 + μ (2,j) b 2 for j 0 where μ (1,0) and μ (2,0) are determined from ŵ (0), μ (1,j) = γ 1 j 1, and γ 1 is a real constant. The values γ 1 and μ (2,j) for j 1 are determined according to the following cases: 1. If K 2 b 2 = 0, then μ (2,j) = 0 for all j 1 and γ 1 = μ (1,0). 2. If K 2 b 2 = γ 3 b 1 for some non-zero γ 3 R, then μ (2,j) = 0 for all j 1, b 1 is an eigenvector of K 2, and γ 1 = μ (1,0) + 2μ (2,0) γ 3 /b T 1 K 2b If b T 2 K 2b 2 = 0 and ρ 1, then μ (2,j) = 0 for all j 1, K 2 b 2 = γ 3 K 2 b 1 for some γ 3 R, and γ 1 = μ (1,0) + μ (2,0) γ If b T 2 K 2b 2 = 0 and ρ > 1, then γ 1 = μ (1,0), μ (2,j) = μ (2,0) [sign(ρ)( ρ ρ 2 1)] j for all j 1, and K 2 is non-singular. Theorem 4.5 together with Remark 4.2 describe the behaviour of the boundary layer in case of Dirichlet boundary condition along ˆΓ when C and T C are as in Remark 4.4. The boundary layer is a constant over S; decays in the first cell and then continues as a constant over the rest of S; or decays exponentially. In the latter case, the amplitude and the decay rate ( ρ ρ 2 1) depend on μ (2,0) and ρ, respectively. The decay rate monotonically decreases from 1 (when ρ 1 + ) to 0 (when ρ ). Indeed, the decay rate is (2 + 3) 1 when ρ =2, and lim ρ ρ ( ρ ρ 2 1) = 1 2. Remark 4.6 For general T C, one can follow Example 2 to compute the boundary layer ŵ corresponding to ˆq in S under Neumann boundary conditions. The idea is to assume that ŵ ˆΓ is known and proceed as if we have the Dirichlet boundary conditions. Again, we assume C and T C to be the same as in Remark 4.4. If C is Criss Cross, or one of its distorted versions, then ˆv ˆΓ is either 0 or 1 for any ˆv S bl (S). Hence, and by virtue of the argument used in Example 2,

26 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1885 B S ( ˆψ, ˆv) = 0 for all ˆv S bl (S). Since ŵ 0asŷ, ŵ is identically 0. For the other three patterns, or their distorted versions, we use Theorem 4.5 and K 1 ŵ (0) + KT 2 ŵ(2) = B C 1 ( ˆψ, ˆ 2 )b 2 lim ŵ = 0 (37) ŷ where ˆ 2 is the Lagrange basis function at z (0,2) and the first part of (37) generalizes (27). 5. ASYMTOTIC BEHAVIOUR OF THE ZZ R IN ATCHES ADJACENT TO Ω We are now ready to evaluate the performance of the ZZ R in patches adjacent to Ω using the tests of Section 3. The following steps sum up what should be done in any of these tests: 1. Choose D, the aspect ratio of C, and T C. 2. For i = 1, 2, 3, compute ˆq per i = Π per C ( ˆq i ˆq i,i ) as explained in Example 1. bl 3. For i = 1, 2, 3, compute ˆq i using Theorem 4.5 and Remark Construct G ˆq i for i = 1, 2, 3. Note that computing G ˆq i in C is somewhat tricky as it should be computed at some nodes attached to C before processing the nodes in C, and the nodes should be processed in a certain order. The reader may find Figure 3 helpful in this aspect. 5. Finally, compute M e and M a using their definitions in Section 3. In the following tests, we will consider the effects of mesh distortion, cell aspect ratio, and material properties. As we have seen in Section 3, there is a slight difference when the tests are carried out for the class of harmonic quadratic polynomials. Hence, this class will have no special treatment in this section and all the tests are carried out for homogeneous quadratic polynomials. We begin with the results of the distortion tests which are reported in Figures A glimpse over these figures reveals the following points. For distortions within practical ranges, the ZZ R performs better than the ZZ SR regardless of the boundary conditions. In the Regular and the Chevron cases, the ZZ R and the ZZ SR are comparable while the ZZ R does better in both the Union Jack and the Criss Cross cases. The ZZ R performs better in Dirichlet boundary conditions than in Neumann boundary conditions in both the Chevron and the Union Jack cases while the opposite of this happens to the ZZ SR. In the Criss Cross case, the boundary layer is either 0 or constant as was shown in Section 4. Consequently, the boundary layer has no effect on both the ZZ R and the ZZ SR as we switch from internal patches to patches adjacent to Ω. Among all the tests in this section, excessive mesh distortion has the severest effect on the performance of both the ZZ R and the ZZ SR. The aspect ratio of C has very little effect on the performance of both the ZZ R and the ZZ SR as shown in Table II and Figure 24, but the reaction of the ZZ R is slightly

27 1886 Z. ZHANG AND A. NAGA Figure 16. The change in the robustness index as T C is distorted from the Regular pattern. Boundary condition on ˆΓ is of Neumann type. Figure 17. The change in the robustness index as T C is distorted from the Regular pattern. Boundary condition on ˆΓ is of Dirichlet type.

28 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1887 Figure 18. The change in the robustness index as T C is distorted from the Chevron pattern. Boundary condition on ˆΓ is of Neumann type. Figure 19. The change in the robustness index as T C is distorted from the Chevron pattern. Boundary condition on ˆΓ is of Dirichlet type.

29 1888 Z. ZHANG AND A. NAGA Figure 20. The change in the robustness index as T C is distorted from the Union Jack pattern. Boundary condition on ˆΓ is of Neumann type. better. Note that the ZZ R is asymptotically exact regardless of a in three of the four patterns. In the Union Jack pattern, Figure 24(a), the robustness index is almost zero. Meanwhile, the ZZ SR is exact only in the Regular pattern. This is true regardless of the boundary conditions. Finally, the material properties have very little effect on the ZZ R as shown in Table III and Figure 25. The ZZ R is asymptotically exact in three of the four patterns regardless of θ,d, or the boundary conditions. Remark 5.1 The definition of the R-recovered gradient at boundary nodes greatly affects the quality of the ZZ R in patches adjacent to Ω. As in the SR, there is more than one way to define the R-recovered gradient at boundary nodes. Studying the asymptotic quality of the ZZ R is a good tool in determining the best definition as was done with ZZ SR in Reference [12]. The following are some possible ways to define the R-recovered gradient at boundary. One way is to treat boundary nodes as in the SR. The disadvantages to this approach are: (1) the incapability of handling nodes that are not attached to internal nodes, and (2) the resulting error indicators are sensitive to cell aspect ratio in the Criss Cross pattern. Another way is to treat the boundary nodes in the same way internal nodes are treated. Of course, this requires constructing patches for the boundary nodes. This can be done in a variety

30 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1889 Figure 21. The change in the robustness index as T C is distorted from the Union Jack pattern. Boundary condition on ˆΓ is of Dirichlet type. Figure 22. The change in the robustness index as T C is distorted from the Criss Cross pattern. Boundary condition on ˆΓ is of Neumann type.

31 1890 Z. ZHANG AND A. NAGA Figure 23. The change in the robustness index as T C is distorted from the Criss Cross pattern. Boundary condition on ˆΓ is of Dirichlet type. Table II. Response of λ min and λ Max to change in a boundary patches. Dirichlet boundary conditions Neumann boundary conditions attern R SR R SR Regular λ min = 1 λ min = 1 λ min = 1 λ min = 1 λ max = 1 λ max = 1 λ max = 1 λ max = 1 Chevron λ min = 1 λ min = 50a (a 2 +1) See Figure 24(c) See Figure 24(c) λ max = 1 λ max = Union Jack See Figure 24(a) See Figure 24(a) λ min = 1 See Figure 24(d) λ max = 1 Criss Cross λ min = 1 See Figure 24(b) λ min = 1 See Figure 24(e) λ max = 1 λ max = 1 of ways as explained in References [6, 11]. Unfortunately, the resulting ZZ R is sensitive to cell aspect ratio in the Criss Cross pattern. However, in both of these two approaches, the ZZ R is more robust than the ZZ SR both under mesh distortion and under changes in material properties.

32 VALIDATION OF THE A OSTERIORI ERROR ESTIMATOR 1891 Figure 24. The change in the robustness index as the aspect ratio of C changes when the mesh pattern in C is the Union Jack or the Criss Cross. For the other two patterns, see Table II: (a) Union Jack, Dirichlet; (b) Criss Cross, Dirichlet; (c) Chevron, Neumann; (d) Union Jack, Neumann; and (e) Criss Cross, Neumann. Table III. Response of λ min and λ max to change in θ when d = 100 boundary patches. Dirichlet boundary conditions Neumann boundary conditions attern R SR R SR Regular λ min = 1 λ min = 1 λ min = 1 λ min = 1 λ max = 1 λ max = 1 λ max = 1 λ max = 1 Chevron λ min = 1 λ min = See Figure 25(c) See Figure 25(c) λ max = 1 λ max = Union Jack See Figure 25(a) See Figure 25(a) λ min = 1 See Figure 25(d) λ max = 1 Criss Cross λ min = 1 See Figure 25(b) λ min = 1 See Figure 25(e) λ max = 1 λ max = 1 In conclusion, it was shown that the ZZ R is as good as or better than the ZZ SR. It seems that the material properties and the cell aspect ratio have little effect on the robustness of both of the two error estimators while the mesh geometry has the largest effect.

33 1892 Z. ZHANG AND A. NAGA Figure 25. The change in the robustness index as the material orthotropy orientation angle θ changes from 90 to 90 when d = 100. The mesh pattern in C is the Union Jack or the Criss Cross. For the other two patterns, see Table III: (a) Union Jack, Dirichlet; (b) Criss Cross, Dirichlet; (c) Chevron, Neumann; (d) Union Jack, Neumann; and (e) Criss Cross, Neumann. ACKNOWLEDGEMENTS The authors would like to thank rofessor I. Babuška for the fruitful discussion that led to the proof of roposition 4.1. REFERENCES 1. Bank RE, Weiser A. Some a posteriori error estimators for elliptic partial differential equations. Mathematics of Computation 1985; 44: Zienkiewicz OC, Zhu JZ. A simple error estimator and adaptive procedure for practical engineering analysis. International Journal for Numerical Methods in Engineering 1987; 24: Li XD, Wiberg N-E. A posteriori error estimate by element patch postprocessing, adaptive analysis in energy norm and L 2 norms. Computers and Structures 1994; 53(4): Wiberg N-E, Li XD. Superconvergent patch recovery of finite element solution and a posteriori error L 2 norm estimate. Communications in Numerical Methods in Engineering 1994; 10(4): Zienkiewicz OC, Zhu JZ. The superconvergence patch recovery and a posteriori error estimates, part I: the recovery technique. International Journal for Numerical Methods in Engineering 1992; 33: Zhang Z, Naga A. A new finite element gradient recovery method: superconvergence property. SIAM Journal on Scientific Computing, accepted for publication. 7. Ainsworth M, Oden JT. A osteriori Error Estimation in Finite Element Analysis. Wiley Interscience: New York, 2000; Babuška I, Strouboulis T. The Finite Element Method and its Reliability. Oxford University ress: London, 2001.

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