Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem


Authors: Eugene O'Riordan and Martin Stynes
Journal: Math. Comp. 47 (1986), 555-570
MSC: Primary 65L10
DOI: https://doi.org/10.1090/S0025-5718-1986-0856702-7
MathSciNet review: 856702
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A finite-element method with exponential basis elements is applied to a selfadjoint, singularly perturbed, two-point boundary value problem. The tridiagonal difference scheme generated is shown to be uniformly second-order accurate for this problem (i.e., the nodal errors are bounded by $ C{h^2}$, where C is independent of the mesh size h and the perturbation parameter). With a certain choice of trial functions, uniform first-order accuracy is obtained in $ {L^\infty }[0,1]$.


References [Enhancements On Off] (What's this?)

  • [1] I. P. Boglaev, "A variational difference scheme for a boundary value problem with a small parameter in the highest derivative," U.S.S.R. Comput. Math. and Math. Phys., v. 21, 1981, no. 4, pp. 71-81. MR 630072 (83f:65122)
  • [2] E. P. Doolan, J. J. H. Miller & W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980. MR 610605 (82h:65053)
  • [3] 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., v. 7, 1981, pp. 3-15. MR 611944 (82d:65059)
  • [4] A. F. Hegarty, J. J. H. Miller & E. O'Riordan, "Uniform second order difference schemes for singular perturbation problems," Boundary and Interior Layers--Computational and Asymptotic Methods (J. J. H. Miller, ed.), Boole Press, Dublin, 1980, pp. 301-305. MR 589380 (83h:65095)
  • [5] P. W. Hemker, A Numerical Study of Stiff Two-point Boundary Value Problems, Mathematical Centre Tracts, No. 80, Amsterdam, 1977. MR 0488784 (58:8294)
  • [6] J. J. H. Miller, "On the convergence, uniformly in $ \varepsilon $, of difference schemes for a two-point boundary value singular perturbation problem," Numerical Analysis of Singular Perturbation Problems (P. W. Hemker & J. J. H. Miller, eds.), Academic Press, New York, 1979, pp. 467-474. MR 556537 (81f:65061)
  • [7] K. Niijima, "On a three-point difference scheme for a singular perturbation problem without a first derivative term I," Mem. Numer. Math., v. 7, 1980, pp. 1-10. MR 588462 (82a:65059)
  • [8] M. A. Protter & H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, N. J., 1967. MR 0219861 (36:2935)
  • [9] E. O'Riordan, Finite Element Methods for Singularly Perturbed Problems, Ph.D. thesis, School of Mathematics, Trinity College, Dublin, 1982.
  • [10] A. H. Schatz & L. B. Wahlbin, "On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions," Math. Comp., v. 40, 1983, pp. 47-89. MR 679434 (84c:65137)
  • [11] G. I. Shishkin, "A difference scheme on a non-uniform mesh for a differential equation with a small parameter in the highest derivative," U.S.S.R. Comput. Math. and Math. Phys., v. 23, no. 3, 1983, pp. 59-66. MR 706886 (85g:65094)
  • [12] M. Stynes & E. O'Riordan, "A superconvergence result for a singularly perturbed boundary value problem," BAIL III--Proc. Third International Conference on Boundary and Interior Layers (J. J. H. Miller, ed.), Boole Press, Dublin, 1984, pp. 309-313. MR 774624 (86b:65084)
  • [13] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR 0158502 (28:1725)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L10

Retrieve articles in all journals with MSC: 65L10


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1986-0856702-7
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society