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

O’Riordan, Eugene; Stynes, Martin

doi:10.1090/S0025-5718-1986-0856702-7

A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem
HTML articles powered by AMS MathViewer

by Eugene O’Riordan and Martin Stynes PDF

Math. Comp. 47 (1986), 555-570 Request permission

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

I. P. Boglaev, A variational difference scheme for a boundary value problem with a small parameter multiplying the highest derivative, Zh. Vychisl. Mat. i Mat. Fiz. 21 (1981), no. 4, 887–896, 1069 (Russian). MR 630072
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
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, J. J. H. Miller, and E. O’Riordan, Uniform second order difference schemes for singular perturbation problems, Boundary and interior layers—computational and asymptotic methods (Proc. Conf., Trinity College, Dublin, 1980) Boole, Dún Laoghaire, 1980, pp. 301–305. MR 589380
P. W. Hemker, A numerical study of stiff two-point boundary problems, Mathematical Centre Tracts, No. 80, Mathematisch Centrum, Amsterdam, 1977. MR 0488784
John J. H. Miller, On the convergence, uniformly in $\varepsilon$, of difference schemes for a two point boundary singular perturbation problem, Numerical analysis of singular perturbation problems (Proc. Conf., Math. Inst., Catholic Univ., Nijmegen, 1978) Academic Press, London-New York, 1979, pp. 467–474. MR 556537
Koichi Niijima, On a three-point difference scheme for a singular perturbation problem without a first derivative term. I, II, Mem. Numer. Math. 7 (1980), 1–27. MR 588462
Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861

Finite Element Methods for Singularly Perturbed Problems

A. H. Schatz and L. B. Wahlbin, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. Comp. 40 (1983), no. 161, 47–89. MR 679434, DOI 10.1090/S0025-5718-1983-0679434-4
G. I. Shishkin, Difference scheme on a nonuniform grid for a differential equation with small parameter multiplying the highest derivative, Zh. Vychisl. Mat. i Mat. Fiz. 23 (1983), no. 3, 609–619 (Russian). MR 706886
Martin Stynes and Eugene O’Riordan, A superconvergence result for a singularly perturbed boundary value problem, BAIL III (Dublin, 1984) Boole Press Conf. Ser., vol. 6, Boole, Dún Laoghaire, 1984, pp. 309–313. MR 774624
Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502