A uniformly convergent method for a singularly perturbed semilinear reaction–diffusion problem with multiple solutions

Sun, Guangfu; Stynes, Martin

doi:10.1090/S0025-5718-96-00753-3

A uniformly convergent method for a singularly perturbed semilinear reaction–diffusion problem with multiple solutions
HTML articles powered by AMS MathViewer

by Guangfu Sun and Martin Stynes PDF

Math. Comp. 65 (1996), 1085-1109 Request permission

Abstract:

This paper considers a simple central difference scheme for a singularly perturbed semilinear reaction–diffusion problem, which may have multiple solutions. Asymptotic properties of solutions to this problem are discussed and analyzed. To compute accurate approximations to these solutions, we consider a piecewise equidistant mesh of Shishkin type, which contains $O(N)$ points. On such a mesh, we prove existence of a solution to the discretization and show that it is accurate of order $N^{-2}\ln ^2 N$, in the discrete maximum norm, where the constant factor in this error estimate is independent of the perturbation parameter $\varepsilon$ and $N$. Numerical results are presented that verify this rate of convergence.

References

K. W. Chang and F. A. Howes, Nonlinear singular perturbation phenomena: theory and applications, Applied Mathematical Sciences, vol. 56, Springer-Verlag, New York, 1984. MR 764395, DOI 10.1007/978-1-4612-1114-3
C.M. D’Annunzio, Numerical analysis of a singular perturbation problem with multiple solutions, Ph.D. Dissertation, University of Maryland at College Park, 1986 (unpublished).
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
Paul C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal. 52 (1973), 205–232. MR 374665, DOI 10.1007/BF00247733
Eugene C. Gartland Jr., Graded-mesh difference schemes for singularly perturbed two-point boundary value problems, Math. Comp. 51 (1988), no. 184, 631–657. MR 935072, DOI 10.1090/S0025-5718-1988-0935072-1
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
Dragoslav Herceg, Uniform fourth order difference scheme for a singular perturbation problem, Numer. Math. 56 (1990), no. 7, 675–693. MR 1031441, DOI 10.1007/BF01405196
Dragoslav Herceg and Nenad Petrović, On numerical solution of a singularly perturbed boundary value problem. II, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 17 (1987), no. 1, 163–186 (English, with Serbo-Croatian summary). MR 939306
Jens Lorenz, Stability and monotonicity properties of stiff quasilinear boundary problems, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 12 (1982), 151–175 (English, with Serbo-Croatian summary). MR 735755
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
Robert E. O’Malley Jr., Singular perturbation methods for ordinary differential equations, Applied Mathematical Sciences, vol. 89, Springer-Verlag, New York, 1991. MR 1123483, DOI 10.1007/978-1-4612-0977-5
Eugene O’Riordan and Martin Stynes, A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem, Math. Comp. 47 (1986), no. 176, 555–570. MR 856702, DOI 10.1090/S0025-5718-1986-0856702-7
J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
H.-G. Roos, Global uniformly convergent schemes for a singularly perturbed boundary-value problem using patched base spline-functions, J. Comput. Appl. Math. 29 (1990), no. 1, 69–77. MR 1032678, DOI 10.1016/0377-0427(90)90196-7
G. I. Shishkin, Grid approximation of singularly perturbed parabolic equations with internal layers, Soviet J. Numer. Anal. Math. Modelling 3 (1988), no. 5, 393–407. Translated from the Russian. MR 974091, DOI 10.1515/rnam.1988.3.5.393
Donald R. Smith, Singular-perturbation theory, Cambridge University Press, Cambridge, 1985. An introduction with applications. MR 812466
Relja Vulanović, On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 13 (1983), 187–201 (English, with Serbo-Croatian summary). MR 786443
R. Vulanović, Exponential fitting and special meshes for solving singularly perturbed problems, IV Conference on Applied Mathematics (B. Vrdoljak, ed.) Split, 1985, pp.59–64.