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

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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 , 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 .

**[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 , 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)**

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0856702-7

Article copyright:
© Copyright 1986
American Mathematical Society