## 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**
T. M. El-Mistikawy & M. J. Werle, "Numerical method for boundary layers with blowing—the exponential box scheme," - 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
E. C. Gartland, Jr., - 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**
T. Kato, - 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**
D. R. Smith, - 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

*AIAA J.*, v. 16, 1978, pp. 749-751. E. C. Gartland, Jr.,

*Strong Stability and a Representation Result for a Singular Perturbation Problem*, Technical Report AMS 87-1, Dept. of Mathematics, Southern Methodist University, January, 1987.

*An Analysis of the Allen-Southwell Finite-Difference Scheme for a Model Singular Perturbation Problem*, Technical Report AMS 87-2, Dept. of Mathematics, Southern Methodist University, April, 1987.

*Perturbation Theory for Linear Operators*, 2nd ed., Springer-Verlag, Berlin, 1980.

*A Green Function for a Singularly Perturbed Dirichlet Problem*, Technical Report, Dept. of Mathematics, University of California, San Diego, March, 1984.

## Additional Information

- © Copyright 1988 American Mathematical Society
- Journal: Math. Comp.
**51**(1988), 93-106 - MSC: Primary 65L10; Secondary 65L60
- DOI: https://doi.org/10.1090/S0025-5718-1988-0942145-6
- MathSciNet review: 942145