A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions

Authors:
Eugene O’Riordan and Martin Stynes

Journal:
Math. Comp. **57** (1991), 47-62

MSC:
Primary 65N30; Secondary 35B25, 35B45

DOI:
https://doi.org/10.1090/S0025-5718-1991-1079029-1

MathSciNet review:
1079029

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Abstract: We analyze a new Galerkin finite element method for numerically solving a linear convection-dominated convection-diffusion problem in two dimensions. The method is shown to be convergent, uniformly in the perturbation parameter, of order in a global energy norm which is stronger than the norm. This order is optimal in this norm for our choice of trial functions.

**[1]**E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders,*Uniform numerical methods for problems with initial and boundary layers*, Boole Press, Dún Laoghaire, 1980. MR**610605****[2]**K. V. Emel'janov,*A difference scheme for a three dimensional elliptic equation with a small parameter multiplying the highest derivative*, Boundary Value Problems for Equations of Mathematical Physics, Ural Scientific Center, U.S.S.R. Academy of Sciences, 1973, pp. 30-42.**[3]**Eugene C. Gartland Jr.,*Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem*, Math. Comp.**48**(1987), no. 178, 551–564, S5–S9. MR**878690**, https://doi.org/10.1090/S0025-5718-1987-0878690-0**[4]**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. MR**0473443****[5]**A. F. Hegarty,*Analysis of finite difference methods for two-dimensional elliptic singular perturbation problems*, Ph.D. thesis, Trinity College, Dublin, 1986.**[6]**A. F. Hegarty, E. O'Riordan, and M. Stynes,*A comparison of uniformly convergent difference schemes for two-dimensional convection-diffusion problems*(submitted for publication).**[7]**A. M. Il′in,*A difference scheme for a differential equation with a small parameter multiplying the highest derivative*, Mat. Zametki**6**(1969), 237–248 (Russian). MR**0260195****[8]**Claes Johnson, Uno Nävert, and Juhani Pitkäranta,*Finite element methods for linear hyperbolic problems*, Comput. Methods Appl. Mech. Engrg.**45**(1984), no. 1-3, 285–312. MR**759811**, https://doi.org/10.1016/0045-7825(84)90158-0**[9]**C. Johnson, A. H. Schatz, and L. B. Wahlbin,*Crosswind smear and pointwise errors in streamline diffusion finite element methods*, Math. Comp.**49**(1987), no. 179, 25–38. MR**890252**, https://doi.org/10.1090/S0025-5718-1987-0890252-8**[10]**Olga A. Ladyzhenskaya and Nina N. Ural’tseva,*Linear and quasilinear elliptic equations*, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR**0244627****[11]**Koichi Niijima,*Pointwise error estimates for a streamline diffusion finite element scheme*, Numer. Math.**56**(1990), no. 7, 707–719. MR**1031443**, https://doi.org/10.1007/BF01405198**[12]**E. O'Riordan and M. Stynes,*Analysis of difference schemes for singularly perturbed differential equations using a discretized Green's function*, BAIL IV, Proc. Fourth Internat. Conf. on Interior and Boundary Layers (J. J. H. Miller, ed.), Boole Press, Dublin, 1986, pp. 157-168.**[13]**-,*A uniformly convergent finite difference scheme for an elliptic singular perturbation problem*, Discretization Methods of Singular Perturbations and Flow Problems (L. Tobiska, ed.), Technical University "Otto von Guericke" Magdeburg, 1989, pp. 48-55.**[14]**H.-G. Roos,*Necessary convergence conditions for upwind schemes in the two-dimensional case*, Internat. J. Numer. Methods Engrg.**21**(1985), no. 8, 1459–1469. MR**799066**, https://doi.org/10.1002/nme.1620210808**[15]**Shagi-Di Shih and R. Bruce Kellogg,*Asymptotic analysis of a singular perturbation problem*, SIAM J. Math. Anal.**18**(1987), no. 5, 1467–1511. MR**902346**, https://doi.org/10.1137/0518107**[16]**Martin Stynes and Eugene O’Riordan,*Finite element methods for elliptic convection-diffusion problems*, BAIL V (Shanghai, 1988) Boole Press Conf. Ser., vol. 12, Boole, Dún Laoghaire, 1988, pp. 65–76. MR**990253****[17]**E. A. Volkov,*Differentiability properties of solutions of boundary value problems for the Laplace and Poisson equations*, Proc. Steklov Inst. Math.**77**(1965), pp. 101-126.

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DOI:
https://doi.org/10.1090/S0025-5718-1991-1079029-1

Article copyright:
© Copyright 1991
American Mathematical Society