Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers

Transcription

1 Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers S. Franz, T. Linß, H.-G. Roos Institut für Numerische Mathematik, Technische Universität Dresden, D-01062, Germany Abstract In this paper we analyse the superconvergence property of the streamline diffusion finite element method (SDFEM) in the case of elliptic problems with characteristic layers. For the SDFEM we give optimal parameter choices for maximal stability in the induced streamline diffusion norm. It is shown that inside the parabolic boundary layer the SDFEM-parameter can be chosen of order ε 1/4 N 2 which is also confirmed by numerical results. Key words: superconvergence, convection-diffusion problems, streamline-diffusion method, numerical analysis, characteristic layers 1991 MSC: 65N12, 65N30, 65N50 1 Introduction Consider the model convection-diffusion equation Lu := ε u bu x + cu = f in Ω = (0, 1) 2, (1.1a) subject to Dirichlet boundary conditions u = 0 on Γ = Ω (1.1b) with b W 1 (Ω), c L (Ω), b β on Ω with a positive constant β. Here 0 < ε 1 is a small perturbation parameter whose presence gives rise to addresses: sebastian.franz@tu-dresden.de (S. Franz), torsten.linss@tu-dresden.de (T. Linß), hans-goerg.roos@tu-dresden.de (H.-G. Roos). Preprint submitted to Elsevier Science 8 October 2007

2 an exponential layer of width O (ε) near the outflow boundary at x = 0 and to two parabolic layers of width O ( ε) near the characteristic boundaries at y = 0 and y = 1; see Fig x y Fig. 1. Typical solution to (1.1) with parabolic layers (left and right) and an exponential layer (front) Furthermore we shall assume that c b x γ > 0 (1.2) which ensures the coercivity of the bilinear form associated with the differential operator L. Therefore (1.1) possesses a unique solution in H 1 0(Ω). Note that (1.2) can always be ensured by a simple transformation ũ(x, y) = u(x, y) e κx with κ chosen appropriately. Because of the presence of layers the use of quasi uniform meshes does not give accurate approximations of (1.1) unless the mesh size is of the size of the perturbation parameter which in practice constitutes a prohibitive restriction. Therefore layer-adapted meshes have to be used to obtain efficient discretizations. Based on a priori knowledge of the layer behaviour we shall construct piecewise uniform meshes so called Shishkin meshes that resolve the layers and yield robust (or uniform) convergence with respect to the perturbation parameter. On these meshes two finite element discretizations will be analysed: the standard Galerkin FEM (GFEM) and the streamline-diffusion FEM (SDFEM) first proposed by Hughes and Brooks [7] both using bilinear trial and test functions. The present study focuses on the SDFEM. For problems of type (1.1) with only exponential layers both methods on Shishkin meshes are well understood. For the Galerkin method uniform convergence of almost first order in the energy norm was established by O Riordan 2

3 and Stynes [17], while Zhang [19] and Linß [12] proved uniform superconvergence of almost second order in discrete versions of that norm. The SDFEM was studied by Stynes and Tobiska [18] who derived uniform superconvergence in the streamline-diffusion norm of almost second order. In the present paper problems with parabolic (or characteristic) layers will be considered. Practically these problems are very important. They can be considered as models of the flow past a surface. Particular attention will be paid to the choice of the streamline-diffusion parameter inside the parabolic layers where the mesh is aligned to the flow and anisotropically refined. It was observed [10,13] that when the stabilization parameter is chosen according to standard recommendations [4, p. 132] proportional to the streamline diameter of the mesh cell the accuracy is adversely affected. An alternative and significantly smaller choice based on residual free bubbles is advertised in [13]. However the argument is not rigorous and uses heuristic arguments. Here we shall give for the first time a rigorous analysis of the SDFEM on Shishkin meshes for convection-diffusion problems with parabolic layers. Uniform superconvergence of almost second order is established. Simultaneously the issue of choosing the stabilization parameter in the SDFEM is addressed. The optimal choice inside the parabolic layers turns out to be proportional to ε 1/4 h 2 sd, where h sd is the streamline diameter of the relevant mesh cells. The analysis for parabolic boundary layers presented here can be considered as a first step towards the treatment of problems with more practical relevance. These are often characterized by the existence of inner parabolic layers, see, e.g., Piet Hemker s well known example in [6]. The paper is organized as follows. In the first Section piecewise uniform layeradapted meshes are constructed based on a priori knowledge of the layer behaviour of the solution. Sections 3 and 4 quote results from [5] for the interpolation error and the Galerkin FEM. Section 5 is devoted to the convergence and superconvergence analysis of the SDFEM. Finally, numerical experiments that illustrate our theoretical results are presented in Section 6. NB. The analysis contains some very technical details. These are deferred to the appendix. In doing so we are attempting to provide a presentation of the essential ideas and results that is easier to access. Notation. Throughout C denotes a generic constant that is independent of both the perturbation parameter ε and the number of mesh points used. 2 Solution decomposition and layer-adapted meshes As mentioned before the solution u of (1.1) possesses an exponential layer at x = 0 and two parabolic layers at y = 0 and y = 1. For our later analysis we shall suppose that u can be split into a regular solution component and various layer parts: 3

4 Assumption 1 The solution u of (1.1) can be decomposed as u = v + w 1 + w 2 + w 12, where for all x, y [0, 1] and 0 i + j 2 we have the pointwise estimates x i yv(x, j y) C, x i yw j 2 (x, y) x i yw j 1 (x, y) Cε i e βx/ε, Cε ( j/2 e y/ ε + e ) (1 y)/ ε, x i yw j 12 (x, y) Cε (i+j/2) e ( βx/ε e y/ ε + e ) (1 y)/ ε (2.1a) (2.1b) (2.1c) and for 0 i + j 3 the L 2 bounds x i yv j C, i 0,Ω x yw j 0,Ω 1 Cε i+1/2, x i yw j 0,Ω 2 Cε j/2+1/4, x i yw j 0,Ω 12 Cε i j/2+3/4 (2.2a) (2.2b) and x i yu j C ( 1 + ε i+1/2 + ε j/2+1/4 + ε i j/2+3/4). 0,Ω (2.2c) Remark 2 For i + j 2 the L 2 bounds (2.2) follow from the pointwise bounds (2.1). For constant coefficients Kellogg and Stynes [8,9] give sufficient compatibility conditions on the data that ensure the existence of the decomposition (2.1) and (2.2); for a more detailed discussion see [5]. When discretizing (1.1), we use a piecewise uniform mesh a so-called Shishkin mesh with N mesh intervals in both x- and y-direction which condenses in the layer regions. For this purpose define the mesh transition parameters λ x := min { 1 2, σε } β ln N { } 1 and λ y := min 4, σ ε ln N with some user-chosen positive parameter σ that will be fixed later. The domain Ω is divided into four (six) subregions with Ω 12 covering the exponential layer, Ω 21 the parabolic layer and Ω 22 the corner layer, while Ω 11 covers the remaining region where there are no layers, see Fig. 2. These subdomains will be uniformly dissected to give the final triangulation. For the mere sake of simplicity in our subsequent analysis we shall assume that λ x = σε β ln N 1 2 and λ y = σ ε ln N 1 4 (2.3) as is typically the case for (1.1). The mesh transition parameters λ x and λ y have been chosen such that the 4

5 Ω 22 Ω 21 Ω 12 Ω 11 Ω 11 := [λ x, 1] [λ y, 1 λ y ] Ω 12 := [0, λ x ] [λ y, 1 λ y ] Ω 21 := [λ x, 1] ([0, λ y ] [1 λ y, 1]) Ω 22 := [0, λ x ] ([0, λ y ] [1 λ y, 1]) Ω 22 Ω 21 Fig. 2. Subregions of Ω layer terms are of order N σ on Ω 11, i.e., w 1 (x, y) + w 2 (x, y) + w 12 (x, y) CN σ for (x, y) Ω 11. Typically σ is chosen equal to the formal order of the method or to accommodate the error analysis: We shall require σ 5/2 for technical reasons. In order to construct our final mesh let N be divisible by 4. Divide each of the intervals [0, λ x ] and [λ x, 1] uniformly into N/2 subintervals. We get our mesh in x-direction: x i, i = 0,..., N. The mesh in y-direction, y j, j = 0,..., N, is obtained by uniformly dividing [0, λ y ] and [1 λ y, 1] into N/4 subintervals and [λ y, 1 λ y ] into N/2 subintervals. By drawing lines through these mesh points parallel to the x-axis and y-axis the domain Ω is partitioned into rectangles. This triangulation is denoted by T N ; see Fig. 3. The mesh sizes h i := x i x i 1 and k j := y j y j 1 satisfy h i = 2 := 2σε β Fig. 3. Triangulation T N of Ω ln N, for i = 1,..., N/2, N 1 := 2(1 λ x), for i = N/2 + 1,..., N N 5

6 and k j = k 2 := 4σ ε ln N, for j = 1,..., N/4 and j = 3N/4 + 1,..., N N k 1 := 2(1 2λ y), for i = N/4 + 1,..., 3N/4. N The mesh sizes and k satisfy 2 N 1 1 2N 1 and k 2 N 1 k 1 2N 1, (2.4) properties which are essential when inverse inequalities are applied in our later analysis. For the mesh elements we shall use two notations: τ ij = [x i 1, x i ] [y j 1, y j ] for a specific element, and τ for a generic mesh rectangle with base h and height k. 3 The interpolation error Problem (1.1) will be discretized by means of finite elements. Their analysis requires precise knowledge of the interpolation error which will be provided in this section. Given any u C 0 ( Ω) and a triangulation T N of Ω into rectangles we denote by u I the nodal piecewise bilinear interpolant to u over T N. The main interpolation-error results can be obtained using the technique in [3,17]. The adaptation is straightforward, though tedious. Let 1,D and 0,D be the standard (semi-)norms in H 1 (D) and L 2 (D). If D = Ω we drop D from the notation. Set v 2 ε := ε v γ v 2 0 for all v H 1 0(Ω), Theorem 3 The interpolation error on a Shishkin mesh with σ 2 satisfies u u I L (Ω 11 ) CN 2, u u I L (Ω\Ω 11 ) CN 2 ln 2 N and u u I ε CN 1 ln N. Remark 4 Bounds for the interpolation error in the L 2 norm on the various subdomains of Ω are easily obtained from the L bounds: u u I 0,Ωkl (meas Ω kl ) 1/2 u u I,Ωkl, k, l = 1, 2. 6

7 The proof of Theorem 3 makes use of the following anisotropic interpolation error bounds given in [1, Theorem 2.7] w w I Lp(τ) C { h 2 w xx + hk w Lp(τ) xy + } Lp(τ) k2 w yy Lp(τ) and (w w I ) x Lp(τ) C { h w xx + k w } Lp(τ) xy Lp(τ) (3.1) which hold true for p [1, ] and arbitrary w Wp 2 (Ω). Furthermore ( ) x w w I w L (τ) x + w I L (τ) x L (τ) 2 w x L (τ) is used. Clearly analogous results hold true for (w w I ) y. 4 Galerkin FEM The variational formulation of (1.1) is: Find u H 1 0(Ω) such that a Gal (u, v) := ε( u, v) (bu x, v) + (cu, v) = f(v) := (f, v) for all v H 1 0(Ω), (4.1) where (, ) D denotes the standard scalar product in L 2 (D). If D = Ω we drop the Ω from the notation. Because of (1.2) the bilinear form a Gal (, ) is coercive with respect to the ε-weighted energy norm, i.e., a Gal (v, v) v 2 ε for all v H 1 0(Ω). Thus the Lax-Milgram Lemma ensures the existence of a unique solution u H 1 0(Ω). Let V N H 1 0(Ω) be the finite-element space of piecewise bilinear elements on the Shishkin mesh. Our discretization is: Find u N V N such that a Gal (u N, v N ) = f(v N ) for all v N V N. (4.2) The uniqueness of this solution is guaranteed by the coercivity of a Gal. According to [15], we have for the GFEM on Shishkin meshes applied to (1.1) u u N ε CN 1 ln N if σ 2, 7

8 a result that also holds when linear or mixed linear/bilinear elements are used in the discretization. Moreover we have the superconvergence result [5] u I u N ε CN 2 ln 2 N if σ 5/2, which in contrast is valid only for bilinear elements, but not linear ones. 5 Streamline-diffusion FEM The streamline-diffusion FEM adds weighted residuals to the standard GFEM in order to stabilize the discretization: a Gal (u, v) + τ T N δ τ (f Lu, bv x ) τ = f(v), (5.1) where δ 0 is a user chosen parameter. This modification is consistent with (1.1), i.e., its solution solves (5.1) too. Our discretization reads: Find u N V N such that a SD (u N, v N ) := a Gal (u N, v N ) + a stab (u N, v N ) = f SD (v N ) for all v N V N, with a stab (u, v) := τ T N δ τ ( ε( u, bvx ) τ + (bu x, bv x ) τ (cu, bv x ) τ ) and f SD (u, v) := f(v) τ T N δ τ (f, bv x ) τ. The SDFEM satisfies the orthogonality condition a SD (u u N, v N ) = 0 for all v V N and is coercive with respect to the streamline diffusion norm v 2 SD := v 2 ε + δ τ (bv x, bv x ) τ. τ T N It is shown in, e.g., [14, III.3.2.1] that if 0 δ τ and δ τ c 2 L (τ) γ for all τ T N (5.2) 8

9 then a SD (v, v) 1 2 v 2 SD v V N. We remark that v ε v SD for all v H 1 0(Ω). Thus a SD (, ) enjoys a stronger stability then a Gal (, ). Roughly speaking, the larger δ the more stability is introduced into the method. Our error analysis starts from the coercivity and Galerkin orthogonality: 1 2 u I u N 2 a ( Gal u I u, u I u ) ( N + a stab u I u, u I u ) N. (5.3) SD For the first term on the right-hand side we have on Shishkin meshes with σ 5/2 ( a Gal u I u, χ ) CN 2 ln 2 N χ ε for all χ V N ; (5.4) see [5]. It remains to bound the second term in (5.3) and to analyse the influence of the parameter δ on stability and accuracy. 5.1 Superconvergence and choice of δ The streamline diffusion parameter δ is chosen to be constant on each subdomain of the decomposition of Ω, i.e., we set δ τ = δ τ = δ kl if τ Ω kl, k, l = 1, 2. Theorem 5 Let u N be the streamline-diffusion approximation to u on a Shishkin mesh with σ 5/2. Suppose the stabilization parameter δ satisfies (5.2), δ 12 C εn 2, δ 21 C ε 1/4 N 2, δ 22 C ε 3/4 N 2 and C N 1 if ε N 1, δ 11 C ε 1 N 2 if ε N 1. with some positive constant C independent of ε and the mesh. Then u I u N SD CN 2 ln 2 N. PROOF. Set χ = u I u N and η = u I u. Then by (5.3) and (5.4), we have 1 2 χ 2 SD a stab (η, χ) + CN 2 ln 2 N χ SD. 9

10 On the four subdomains of Ω we have the bounds and a stab (η, χ) Ω11 C ( εδ 11 ln 1/2 N + δ 1/2 11 N σ+1) χ SD, (5.5a) a stab (η, χ) Ω21 Cε 1/4 ( δ 21 + δ 1/2 21 N σ+1) ln 1/2 N χ SD, (5.5b) a stab (η, χ) Ω12 C ( δ 1/2 12 ε 1/2 N σ+1 ln 1/2 N + δ 12 ε 1) χ SD, (5.5c) a stab (η, χ) Ω22 Cδ 22 ε 3/4 ln N χ SD. (5.5d) The proof of Lemma 4.4 in [18] requires only minor modifications to get (5.5a). Details for the other three estimates can be found in the Appendix. Note that our choice of δ 11 implies εδ 11 CN 2 and δ 11 CN 1. The proposition of the theorem follows. Remark 6 (Convergence) Theorems 3 and 5, and the triangle inequality provide bounds for the error in the ε-weighted energy norm: u u N ε CN 1 ln N. Remark 7 Standard recommendation [4] for the stabilization parameter is Ch τ,sd if ε h τ,sd, δ τ = Cε 1 h 2 τ,sd otherwise, where h τ,sd is the diameter of the mesh cell τ in streamline direction. Inside the parabolic layers (on Ω 21 ) this recommendation gives for small ε δ 21 = CN 1, while our analysis states that δ 21 should be chosen at most proportional to N 2, but allowing for some dependence on ε. In [13] the alternative choice δ 21 = CN 2 ln 2 N is derived based on a residualfree bubbles approach and some heuristics. This too implies a significant reduction in the amount of stabilization. 5.2 Bilinear or linear elements? In [16] it was pointed out that in the analysis of the SDFEM on layer-adapted meshes only for some terms the superconvergence properties of bilinear elements are used; for several terms the estimates work both for bilinear and for linear elements. Therefore it is interesting to study what happens if in some of the four subdomains Ω kl bilinear elements are replaced by linear elements and 10

11 in which subdomains the use of bilinears has to be strongly recommended. As a rule of thumb it is favourably to use bilinears in layer regions. (i) First let us study the case that linear elements are used on Ω 11 only. Instead of (5.5a) one can only prove a stab (η, χ) Ω11 C ( δ 11 ε ln 1/2 N + δ 1/2 11 N 1) χ SD, moreover, the Galerkin part also gives only contributions of size O(ε 1/2 N 1 ) and O(N 2 δ 1/2 11 ). However when ε N 1 the standard choice of the parameter δ 11 preserves the superconvergence of order 3/2 for u I u N as it is expected of the SDFEM. (ii) If we use linear elements instead of bilinears in Ω 21, the situation becomes more complicated. First, the inspection of the proof of (5.5b) shows that the estimate of δ 21 ε ( u, bχ x ) Ω21 is independent of superconvergence properties, but the estimate of not. Thus, instead of (5.5b) one gets δ 21 ε (b(u u I ) x, bχ x ) Ω21 a stab (η, χ) Ω21 C ( δ 21 ε 1/4 ln 1/2 N + δ 1/2 21 N 1 + δ 1/2 21 N σ+1 ln 1/2 N ) χ SD. For the Galerkin part we obtain ε( (w 2 w2), I χ) Ω21 Cε 1/4 N 1 ln N and ((w 2 w I 2) x, χ) Ω21 Cε 1/4 N 2 ln 2 Nδ 1/2 21. Thus, for δ 21 = CN 4/3 ln N the stabilization terms and the convective part of Galerkin are of magnitudes O(ε 1/4 N 4/3 ln 3/2 N) and O(N 5/3 ln 1/2 N). Furthermore we have an error term of size O(ε 1/4 N 1 ln N). Therefore, when ε N 1 we have superconvergence of order 5/4 for u I u N. (iii) The worst choice is to use linear elements on Ω 12 Ω 22 ; then it seems only possible to prove u N u I SD CN 1 ln N because nothing better then ε 1/2 w 1 w I 1 CN 1 ln N can be used to estimate the Galerkin part. 11

12 N u u N ε rate u I u N SD rate u u N L rate e e e e e e e e e e e e e e e e e e e e e-04 Table 1 Problem I Convergence It is still necessary to verify the theoretical statements by a detailed numerical study. 6 Numerical Results Two different test problems will be studied. In our experiments the uniform errors are estimated by taking the maximum over various values of ε, i.e. e N = max ε=10 2,10 4,...,10 10 u un. The rate of convergence is estimated by r N = ld(e N /e 2N ). For the computations with the SDFEM we have chosen δ to be the maximal values allowed by Theorem 5 with C = 1. Problem I is given by ε u (2 x)u x u = f in Ω = (0, 1)2, u Ω = 0 with Dirichlet boundary conditions and right-hand side f chosen such that u(x, y) = ( cos πx 2 e x/ε e 1/ε ) ( 1 e ) ( y/ ε 1 e ) (1 y)/ ε 1 e 1/ε 1 e 1/ ε is the exact solution. Table 1 displays the errors in various norms of the SD- FEM for test problem I. They are clear illustrations of the first order convergence result (Remark 6) and the almost second order superconvergence result (Theorem 5). The last column gives the errors in the L norm for which almost second order convergence is also observed, although we do not have theoretical justification for this behaviour. 12

13 ε u u N ε u I u N SD u u N L 1.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-03 Table 2 Problem I Uniformity, N = 256 In Table 2 we verify the dependence of the errors on the perturbation parameter. We have fixed N = 256 and varied the perturbation parameter. We observe ε-independence of the errors in the energy norm and in the SD norm. However we notice an increase of the L error when ε gets smaller. This may be an indication that the stabilization according to Theorem 5 though correct for the SD norm is not optimal for the maximum-norm error. Problem II. ε u u x + u = f in Ω = (0, 1) 2, u Ω = 0 with right-hand side f chosen such that ( 1 e y/ ε ) ( 1 e (1 y)/ ε ) u = sin(πx) 1 e 1/ ε is the solution. This problem has been selected because it exhibits only parabolic layers, the layers we are particularly concerned with. The construction of the Shishkin mesh is changed by using a uniform mesh in x-direction since there is no layer at x = 0. Table 3 gives the errors for the second test problem. The behaviour is similar to Problem I: almost first order convergence in the energy norm and second order superconvergence in the SD norm. However the dependence of the errors on ε is different. For the SD norm we witness a decrease of the errors when ε gets smaller, while the maximum-norm error increase slightly, see Table 4. 13

14 N u u N ε rate u I u N SD rate u u N L rate e e e e e e e e e e e e e e e e e e e e e-04 Table 3 Problem II Convergence ε u u N ε u I u N SD u u N L 1.00e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-03 Table 4 Problem II Uniformity, N = 256 References [1] T. Apel. Anisotropic finite elements: Local estimates and applications. Advances in Numerical Mathematics, Teubner, Stuttgart, 1999 [2] P. Ciarlet. The Finite Element Method for Elliptic Problems, Classics Appl. Math. 40, SIAM, Philadelphia, 2002 [3] M. Dobrowolski and H.-G. Roos. A priori estimates for the solution of convection-diffusion problems and interpolation on Shishkin meshes. Z. Anal. Anwend., 16(4): , 1997 [4] H.C. Elman, D.J. Silvester, A.J. Wathen. Finite Elements and Fast Iterative Solvers. Oxford University Press, 2005 [5] S. Franz, T.Linß. Superconvergence analysis of the Galerkin FEM for a 14

15 singularly perturbed convection-diffusion problem with characteristic layers. 2006, submitted. [6] P.W. Hemker. A singularly perturbed model problem for numerical computation. J. Comput. Appl. Math.,76: , 1996 [7] T. J. R. Hughes and A. Brooks. A multidimensional upwind scheme with no crosswind diffusion. In T. J. R. Hughes, editor, Finite Element Methods for Convection Dominated Flows, pages AMD. Vol. 34, ASME, New York, 1979 [8] R.B. Kellogg, M. Stynes. Corner singularities and boundary layers in a simple convection-diffusion problem. J. Differential Equations, 213: , 2005 [9] R.B. Kellogg, M. Stynes. Sharpened bounds for corner singularities and boundary layers in a simple convection-diffusion problem. Appl. Math. Letters, in press. [10] N. V. Kopteva. How accurate is the streamline-diffusion FEM inside characteristic (boundary and interior) layers? Comput. Meth. Appl. Mech. Engng., 193(45-47): , 2004 [11] Q. Lin. A rectangle test for finite element analysis. In Proceedings of Systems Science & Systems Engineering, pages Great Wall (H.K.) Culture Publish Co., 1991 [12] T. Linß. Uniform superconvergence of a Galerkin finite element method on Shishkin-type meshes. Numer. Methods Partial Differ. Equations, 16(5): , 2000 [13] T. Linß. Anistropic meshes and streamline-diffusion stabilization for convectiondiffusion problems. Commun. Numer. Meth. Engng., 21: , 2005 [14] H.-G. Roos, M. Stynes, L. Tobiska. Numerical Methods for Singularly Perturbed Differential Equations. Springer Series in Computational Mathematics, vol. 24, 1996 [15] H.-G. Roos. Optimal convergence of basic schemes for elliptic boundary value problems with strong parabolic layers. J. Math. Anal. Appl., 267: , 2002 [16] H.-G. Roos. On the streamline-diffusion stabilization for convection-diffusion problems using layer-adapted meshes. Preprint, MATH-NM , 2003 [17] M. Stynes, E. O Riordan. A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem. J. Math. Anal. Appl., 214(1): 36-54, 1997 [18] M. Stynes, L. Tobiska. The SDFEM for a convection-diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy. SIAM J. Numer. Anal., 41(5): , 2003 [19] Zh. Zhang. Finite element superconvergence on Shishkin mesh for 2-d convection-diffusion problems. Math. Comp., 72(243): ,

16 A Appendix, proof of (5.5b-d) Throughout the remaining analysis we shall make frequent use of the following inverse estimates. Let χ be a polynomial on the mesh rectangle τ. Then χ x Lp(τ) Ch 1 χ Lp(τ), χ y Lp(τ) Ck 1 χ Lp(τ) (A.1) and χ Lq(τ) C(meas τ) 1/q 1/p χ Lp(τ) for p, q [1, ], (A.2) where h is the base and k the height of τ. Furthermore we shall employ the following integral identity [11]. Let τ ij T N be an arbitrary mesh rectangle with midpoint ( x i, ỹ j ) and edges parallel to the axes. Then for any function w C 3 ( τ ij ) and any bilinear function χ there holds (w w I ) x χ x = [F j χ x 1 ] τ ij 3 (F 2j ) χ xy w xyy with τ ij F j (y) := (y ỹ j) 2 2 k2 j 8. Application of the Cauchy-Schwarz inequality and the inverse inequality (A.1) gives ( ) (w w I ) x, χ x τ ij Ck2 j w xyy 0,τij χ x 0,τij for all w C 3 ( τ ij ). (A.3) Next note that because of 0 < β b b L (Ω) the two semi norms χ x Lp(D) and bχ x Lp(D) are equivalent with constants independent of ε and the mesh. This fact will be used repeatedly in our analysis without special reference. Furthermore we shall use Proposition 8 Let b W 1 (Ω). Then ( ) b(ϕ ϕ I ) x, bχ x Ω kl C [( l + k ( ) ] k) l ϕ xx 0,Ωkl + k k ϕ xy 0,Ωkl + k 2k ϕ xyy 0,Ωkl χ x 0,Ωkl for k, l = 1, 2. 16

17 PROOF. Let again b ij := b(x i, y j ) and b b ij on τ ij. For an arbitrary rectangle τ we have ( b(ϕ ϕ I ) x, bχ x )τ = τ (b 2 b 2 )(ϕ ϕ I ) x χ x + b 2 τ (ϕ ϕ I ) x χ x The first term on the right-hand side can be bounded by a standard interpolation argument (see [2, Theorem 3.1.5]) and (3.1). For the second term use (A.3). We get ( ) b(ϕ ϕ I ) x, bχ x C [(h+k) ( ) ] h ϕ xx 0,τ +k ϕ xy 0,τ +k 2 ϕ xyy 0,τ χ x 0,τ τ Summing over Ω kl and applying a discrete Cauchy-Schwarz inequality, we are done. (i) Let us first consider Ω 21 which covers the parabolic boundary layers. a stab (u u I, χ) Ω21 = δ 21 { ε ( u, bχ x ) Ω21 + ( b(u u I ) x, bχ x ) Ω 21 ( c(u u I ), bχ x )Ω 21 For the last term we have by Theorem 3 and the Cauchy-Schwarz inequality δ 21 ( c(u u I ), bχ x ) Ω 21 Cδ 21ε 1/4 N 2 ln 5/2 N bχ x 0,Ω21 Cδ 1/2 21 ε 1/4 N 2 ln 5/2 N χ SD. For the second term we proceed as follows. Let w = w 1 + w 12 and w = v + w 2. Then ( ) δ 21 b(w w I ) x, bχ x Ω 21 Cδ 21 ( w x L1(Ω21) χ x L (Ω21) + w Ix ) 0,Ω21 bχ x 0,Ω21 ( Cδ 21 N w x L1 (Ω 21 )ε 1/4 ln 1/2 N χ x 0,Ω21 ) + ε 1/4 ln 1/2 N w I L (Ω21 ) bχ x 0,Ω21 Cδ 1/2 21 ε 1/4 N σ+1 ln 1/2 N χ SD, by (A.1), (A.2) and (2.1). For the term involving w use Proposition 8. δ 21 ( b( w w I ) x, bχ x ) Ω 21 Cδ 21 N [ 2 w xx 0,Ω21 + ε 1/2 ln N w xy 0,Ω21 Cδ 1/2 21 ε 1/4 N 2 ln 2 N χ SD, }. + ε ln 2 N w xyy 0,Ω21 ] χ x 0,Ω21 17

18 When bounding ( u, bχ x ) we use the splitting u = w + w again. δ 21 ε ( w, bχ x ) Ω21 δ21 ε w L1 (Ω 21 ) bχ x L (Ω21 ) by (A.2) and (2.1). Next Cδ 21 εn σ ε 1/2 ln N(Nε 1/4 ln 1/2 ) bχ x 0,Ω21 Cδ 1/2 21 ε 1/4 N σ+1 ln 1/2 N χ SD, and (2.1) imply ( w, bχ x ) Ω21 + ( w, bχ x ) Ω22 = ((b w) x, χ) Ω21 Ω 22 δ 21 ε ( w, bχ x ) Ω21 Cδ21 ε ( ) ε 3/4 χ 0,Ω21 Ω 22 + ε 1/4 ln 1/2 N χ x 0,Ω22 Cδ 21 ε 1/4 ln 1/2 N χ ε. Collecting the above results, we get (5.5b). (ii) The proof of (5.5c), i.e. for the region of the exponential layer, uses precisely the same argument. The only difference is in the bounds for the various (semi-)norms of u and its components. (iii) Finally consider Ω 22. By Theorem 3 we have ( ) c(u u I ), bχ x Ω 22 Cε3/4 N 2 ln 3 N χ x 0. Proposition 8 and a direct calculation of the norms of u involved give ( ) b(u u I ) x, bχ x Ω 22 Cε 1/4 N 2 ln 2 N χ x 0. Finally ε ( u, bχ x ) Ω22 Cε 1/4 ln N χ x 0. This is the dominant term on Ω 22. Remark that χ x 0 ε 1/2 χ SD. We get (5.5d). 18