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Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems

Author: Zhimin Zhang
Journal: Math. Comp. 72 (2003), 1147-1177
MSC (2000): Primary 65N30, 65N15
Published electronically: February 3, 2003
MathSciNet review: 1972731
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Abstract: In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate $O(N^{-2}\ln^2 N + \epsilon N^{-1.5}\ln N)$ in a discrete $\epsilon$-weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter $\epsilon$. Numerical tests indicate that the rate $O(N^{-2}\ln^2 N)$ is sharp for the boundary layer terms. As a by-product, an $\epsilon$-uniform convergence of the same order is obtained for the $L^2$-norm. Furthermore, under the same regularity assumption, an $\epsilon$-uniform convergence of order $N^{-3/2}\ln^{5/2} N + \epsilon N^{-1}\ln^{1/2} N$ in the $L^\infty$ norm is proved for some mesh points in the boundary layer region.

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  • 1. C. Chen and Y. Huang, High Accuracy Theory of Finite Element Methods (in Chinese), Hunan Science Press, P.R. China, 1995.
  • 2. K. Gerdes, J.M. Melenk, D. Schötzau and C. Schwab, The $hp$-Version of the Streamline Diffusion Finite Element Method in Two Space Dimensions, Math. Models Methods Appl. Sci. 11 (2001), 301-337. MR 2001m:65168
  • 3. H. Han and R.B. Kellogg, Differentiability properties of solutions of the equation $-\epsilon\Delta u + ru = f(x,y)$ in a square, SIAM J. Math. Anal. 21 (1990), 394-408. MR 91e:35025
  • 4. C. Johnson, A. Schatz, and L. Wahlbin, Crosswind smear and pointwise errors in the streamline diffusion finite element method, Math. Comp. 49 (1987), 25-38. MR 88i:65130
  • 5. R.B. Kellogg, Boundary layers and corner singularities for a self-adjoint problem, in Boundary Value Problems and Integral Equations in Non-smooth Domains, M. Costabel, M. Dauge, and S. Nicaise eds., Marcel Dekker, New York, 1995, 121-149. MR 95i:35046
  • 6. M. Krízek, P. Neittaanmäki, and R. Stenberg (Eds.), Finite Element Methods: Superconvergence, Post-processing, and A Posteriori Estimates, Lecture Notes in Pure and Applied Mathematics, Vol. 196, Marcel Dekker, Inc., New York, 1998. MR 98i:65003
  • 7. R.D. Lazarov, L. Tobiska, and P.S. Vassilevski, Stream-line diffusion least-squares mixed finite element methods for convection-diffusion problems, East-West J. Numer. Math. 5 (1997), 249-264. MR 98k:76091
  • 8. J. Li and M.F. Wheeler, Uniformly convergent and superconvergence of mixed finite element methods for anisotropically refined grids, SIAM J. Numer. Anal. 38 (2000), 770-798. MR 2001f:65137
  • 9. Q. Lin and N. Yan, Construction and Analysis of High Efficient Finite Elements (in Chinese), Hebei University Press, P.R. China, 1996.
  • 10. T. Linßand M. Stynes, Asymptotic analysis and Shishkin-type decomposition for an elliptic convection-diffusion problem, J. Math. Anal. Appl. 261 (2001), 604-632. MR 2002h:35039
  • 11. J.M. Melenk and C. Schwab, $hp$ FEM for reaction-diffusion equations I: Robust exponential convergence, SIAM J. Numer. Anal. 35 (1998), 1520-1557. MR 99d:65333
  • 12. J.M. Melenk and C. Schwab, Analytic regularity for a singularly perturbed problem, SIAM J. Math. Anal. 30 (1999), 379-40. MR 2000b:35010
  • 13. J.J. Miller, E. O'Riordan, and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996. MR 98c:65002
  • 14. K.W. Morton, Numerical Solution of Convection-Diffusion Problems, Chapman and Hall, London, 1996. MR 98b:65004
  • 15. H.-G. Roos, Layer-adapted grids for singular perturbation problems, ZAMM Z. Angew. Math. Mech. 78 (1998), 291-309. MR 99d:65301
  • 16. H.-G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin, 1996. MR 99a:65134
  • 17. A.H. Schatz, I.H. Sloan, and L.B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal. 33 (1996), 505-521. MR 98f:65112
  • 18. A.H. Schatz and L.B. Wahlbin, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. Comp. 40 (1983), 47-89. MR 84c:65137
  • 19. C. Schwab and M. Suri, The $p$ and $hp$ versions of the finite element method for problems with boundary layers, Math. Comp. 65 (1996), 1403-1429. MR 97a:65067
  • 20. G.I. Shishkin, Discrete approximation of singularly perturbed elliptic and parabolic problems (in Russian), Russian Academy of Sciences, Ural Section, Ekaterinburg, 1992.
  • 21. M. Stynes and E. O'Riordan A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem, J. Math. Anal. Appl. 214 (1997), 36-54. MR 99f:65177
  • 22. L.B. Wahlbin, Local behavior in finite element methods, in Handbook of Numerical Analysis Vol. II, P.G. Ciarlet and J.L. Lions eds., North-Holland Publishing Company, Amsterdam, 1991, 353-522. MR 92f:65001
  • 23. L.B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, Vol. 1605, Springer, Berlin, 1995. MR 98j:65083
  • 24. Z. Zhang, Superconvergent approximation of singularly perturbed problems, Numer. Meth. PDEs 18 (2002), 374-395.
  • 25. G. Zhou, How accurate is the streamline diffusion finite element method? Math. Comp. 66 (1997), 31-44. MR 97f:65171

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Additional Information

Zhimin Zhang
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Keywords: Convection, diffusion, singularly perturbed, boundary layer, Shishkin mesh, finite element method.
Received by editor(s): July 19, 2000
Received by editor(s) in revised form: December 10, 2001
Published electronically: February 3, 2003
Additional Notes: This research was partially supported by the National Science Foundation grants DMS-0074301, DMS-0079743, and INT-0196139
Article copyright: © Copyright 2003 American Mathematical Society

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