An analysis of a uniformly convergent finite difference/finite element scheme for a model singular-perturbation problem

Gartland, Eugene C.

doi:10.1090/S0025-5718-1988-0942145-6

An analysis of a uniformly convergent finite difference/finite element scheme for a model singular-perturbation problem
HTML articles powered by AMS MathViewer

by Eugene C. Gartland PDF

Math. Comp. 51 (1988), 93-106 Request permission

Abstract:

Uniform $\mathcal {O}({h^2})$ convergence is proved for the El-Mistikawy-Werle discretization of the problem $- \varepsilon u”+ au’+ bu = f$ on (0,1), $u(0) = A$, $u(1) = B$, subject only to the conditions $a,b,f \in {\mathcal {W}^{2,\infty }}[0,1]$ and $a(x) > 0, 0 \leq x \leq 1$. The principal tools used are a certain representation result for the solutions of such problems that is due to the author [Math. Comp., v. 48, 1987, pp. 551-564] and the general stability results of Niederdrenk and Yserentant [Numer. Math., v. 41, 1983, pp. 223-253]. Global uniform $\mathcal {O}(h)$ convergence is proved under slightly weaker assumptions for an equivalent Petrov-Galerkin formulation.

References

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
Jim Douglas Jr. and Todd Dupont, Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary problems, Topics in numerical analysis (Proc. Roy. Irish Acad. Conf., University Coll., Dublin, 1972) Academic Press, London, 1973, pp. 89–92. MR 0366044

AIAA J.

Strong Stability and a Representation Result for a Singular Perturbation Problem

Eugene C. Gartland Jr., Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem, Math. Comp. 48 (1987), no. 178, 551–564, S5–S9. MR 878690, DOI 10.1090/S0025-5718-1987-0878690-0

An Analysis of the Allen-Southwell Finite-Difference Scheme for a Model Singular Perturbation Problem

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

Perturbation Theory for Linear Operators

Stephen H. Leventhal, An operator compact implicit method of exponential type, J. Comput. Phys. 46 (1982), no. 1, 138–165. MR 665807, DOI 10.1016/0021-9991(82)90008-0
Jens Lorenz, Stability and consistency analysis of difference methods for singular perturbation problems, Analytical and numerical approaches to asymptotic problems in analysis (Proc. Conf., Univ. Nijmegen, Nijmegen, 1980) North-Holland Math. Stud., vol. 47, North-Holland, Amsterdam-New York, 1981, pp. 141–156. MR 605505
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, An analysis of a superconvergence result for a singularly perturbed boundary value problem, Math. Comp. 46 (1986), no. 173, 81–92. MR 815833, DOI 10.1090/S0025-5718-1986-0815833-8
Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861

A Green Function for a Singularly Perturbed Dirichlet Problem

Donald R. Smith, Singular-perturbation theory, Cambridge University Press, Cambridge, 1985. An introduction with applications. MR 812466
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
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