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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 $ {h^{1/2}}$ in a global energy norm which is stronger than the $ {L^2}$ norm. This order is optimal in this norm for our choice of trial functions.


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  • [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, Dublin, 1980. MR 610605 (82h:65053)
  • [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] E. C. Gartland, Jr., Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem, Math. Comp. 48 (1987), 551-564. MR 878690 (89a:65116)
  • [4] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin, 1977. MR 0473443 (57:13109)
  • [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, Differencing scheme for a differential equation with a small parameter affecting the highest derivative, Mat. Zametki 6 (1969), 237-248; English transl., Math. Notes 6 (1969), 596-602. MR 0260195 (41:4823)
  • [8] C. Johnson, V. Nävert, and I. Pitkäranta, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 45 (1984), 285-312. MR 759811 (86a:65103)
  • [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), 25-38. MR 890252 (88i:65130)
  • [10] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York, 1968. MR 0244627 (39:5941)
  • [11] K. Niijima, Pointwise error estimates for a streamline diffusion finite element scheme, Numer. Math. 56 (1990), 707-719. MR 1031443 (91h:65178)
  • [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), 1459-1469. MR 799066 (86j:65141)
  • [15] S. D. Shih and R. B. Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal. 18 (1987), 1467-1511. MR 902346 (88j:35020)
  • [16] M. Stynes and E. O'Riordan, Finite element methods for elliptic convection-diffusion problems, BAIL V, Proc. Fifth Internat. Conf. on Interior and Boundary Layers (J. J. H. Miller, ed.), Boole Press, Dublin, 1988, pp. 65-76. MR 990253 (90d:65200)
  • [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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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1991-1079029-1
Article copyright: © Copyright 1991 American Mathematical Society

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