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

Abstract

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 ε and N . Numerical results are presented that verify this rate of convergence.

Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem

“…A one-dimensional version of problem (1.1) was studied in [5,17,8]. The present paper extends the analysis [8] to two dimensions.…”

A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem

“…First we want to estimate the expressions containing only the first derivatives in the RHS of inequality (26). From the identity a n −b n = (a−b)(a n−1 +a n−2 b+.…”

“…Now we want to estimate the terms containing the second derivatives from the RHS of (26). Using inequality (19) we get that…”

“…Various different difference schemes have been constructed which are uniformly convergent on equidistant meshes as well as schemes on specially constructed, mostly Shishkin and Bakvhvalov-type meshes, where ϵ-uniform convergence of second order has been demonstrated, see e.g. [11,13,14,24,26,28,29], as well as schemes with ϵ-uniform convergence of order greater than two, see e.g. [7,8,9,32,33].…”

A class of singularly perturbed semilinear differential equations with interior layers

“…In [5] it was shown that numerical methods based on uniform meshes cannot be parameter uniform for semilinear singularly perturbed problems. Sun and Stynes [14] constructed finite difference schemes based on piecewise-uniform meshes for semilinear problems whose solutions exhibit only boundary layer structure. In this paper we are primarily interested in the interior layer behaviour introduced by the discontinuity of f .…”