An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques
HTML articles powered by AMS MathViewer
- by Martin Stynes and Eugene O’Riordan PDF
- Math. Comp. 56 (1991), 663-675 Request permission
Abstract:
We give a new analysis of Petrov-Galerkin finite element methods for solving linear singularly perturbed two-point boundary value problems without turning points. No use is made of finite difference methodology such as discrete maximum principles, nor of asymptotic expansions. On meshes which are either arbitrary or slightly restricted, we derive energy norm and ${L^2}$ norm error bounds. These bounds are uniform in the perturbation parameter. Our proof uses a variation on the classical Aubin-Nitsche argument, which is novel insofar as the ${L^2}$ bound is obtained independently of the energy norm bound.References
- O. Axelsson, Stability and error estimates of Galerkin finite element approximations for convection-diffusion equations, IMA J. Numer. Anal. 1 (1981), no. 3, 329–345. MR 641313, DOI 10.1093/imanum/1.3.329
- Alan E. Berger, Jay M. Solomon, and Melvyn Ciment, An analysis of a uniformly accurate difference method for a singular perturbation problem, Math. Comp. 37 (1981), no. 155, 79–94. MR 616361, DOI 10.1090/S0025-5718-1981-0616361-0
- K. V. Emel′yanov, Difference schemes for singularly perturbed boundary value problems, BAIL IV (Novosibirsk, 1986) Boole Press Conf. Ser., vol. 8, Boole, Dún Laoghaire, 1986, pp. 51–60. MR 926697
- Eugene C. Gartland Jr., An analysis of a uniformly convergent finite difference/finite element scheme for a model singular-perturbation problem, Math. Comp. 51 (1988), no. 183, 93–106. MR 942145, DOI 10.1090/S0025-5718-1988-0942145-6
- Eugene C. Gartland Jr., Graded-mesh difference schemes for singularly perturbed two-point boundary value problems, Math. Comp. 51 (1988), no. 184, 631–657. MR 935072, DOI 10.1090/S0025-5718-1988-0935072-1
- P. P. N. de Groen, A finite element method with a large mesh-width for a stiff two-point boundary value problem, J. Comput. Appl. Math. 7 (1981), no. 1, 3–15. MR 611944, DOI 10.1016/0771-050X(81)90001-2 A. F. Hegarty, E. O’Riordan, and M. Stynes, A comparison of uniformly convergent difference schemes for two-dimensional convection-diffusion problems (in preparation).
- 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 260195
- R. Bruce Kellogg and Alice Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp. 32 (1978), no. 144, 1025–1039. MR 483484, DOI 10.1090/S0025-5718-1978-0483484-9
- K. W. Morton, Galerkin finite element methods and their generalisations, The state of the art in numerical analysis (Birmingham, 1986) Inst. Math. Appl. Conf. Ser. New Ser., vol. 9, Oxford Univ. Press, New York, 1987, pp. 645–680. MR 921681
- K. W. Morton and B. W. Scotney, Petrov-Galerkin methods and diffusion-convection problems in $2$D, The mathematics of finite elements and applications, V (Uxbridge, 1984) Academic Press, London, 1985, pp. 343–366. MR 811047
- Klaus Niederdrenk and Harry Yserentant, Die gleichmäßige Stabilität singulär gestörter diskreter und kontinuierlicher Randwertprobleme, Numer. Math. 41 (1983), no. 2, 223–253 (German, with English summary). MR 703123, DOI 10.1007/BF01390214
- Eugene O’Riordan and Martin Stynes, A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions, Math. Comp. 57 (1991), no. 195, 47–62. MR 1079029, DOI 10.1090/S0025-5718-1991-1079029-1
- Martin Stynes, An adaptive uniformly convergent numerical method for a semilinear singular perturbation problem, SIAM J. Numer. Anal. 26 (1989), no. 2, 442–455. MR 987400, DOI 10.1137/0726025
- Martin Stynes and Eugene O’Riordan, A finite element method for a singularly perturbed boundary value problem, Numer. Math. 50 (1986), no. 1, 1–15. MR 864301, DOI 10.1007/BF01389664
- 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 —, An analysis of a two-point boundary value problem with a boundary layer, using only finite element techniques, Technical Report, Department of Mathematics, University College, Cork, October 1989.
- W. G. Szymczak and I. Babuška, Adaptivity and error estimation for the finite element method applied to convection diffusion problems, SIAM J. Numer. Anal. 21 (1984), no. 5, 910–954. MR 760625, DOI 10.1137/0721059
- M. van Veldhuizen, Higher order schemes of positive type for singular perturbation problems, Numerical analysis of singular perturbation problems (Proc. Conf., Math. Inst., Catholic Univ., Nijmegen, 1978) Academic Press, London-New York, 1979, pp. 361–383. MR 556526
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Math. Comp. 56 (1991), 663-675
- MSC: Primary 65L60; Secondary 34E15
- DOI: https://doi.org/10.1090/S0025-5718-1991-1068809-4
- MathSciNet review: 1068809