Numerical analysis of singularly perturbed nonlinear reaction-diffusion equations

Abstract

This work is concerned with finding accurate numerical approximations to nonlinear reaction-diffusion problems that exhibit layer phenomena. It considers three variations of problems of this type. These are; a two-dimensional steady state equation with Dirichlet boundary conditions exhibiting interior layer solutions; a time-dependent equation with singularly perturbed Neumann boundary conditions with boundary layer solutions; and a steady state equation with singularly perturbed Neumann boundary conditions exhibiting boundary layer solutions. Asymptotic analysis is called upon from previous literature in order to obtain upper and lower solutions to the problems. The theory of Z-fields are then used along with discrete upper and lower solutions to prove existence of a discrete solution and obtain accuracy bounds. Discretisations in the finite difference method and the finite element method are presented on layer adapted meshes such as the Shishkin and Bakhvalov mesh. In cases where incorrect computed solutions are obtained from a conventional discretisation the stabilised method by Kopteva and Savescu [16] is employed, giving solutions of the correct form. It is found that the problems have second-order convergence in space in the maximum norm, with a logarithmic factor for the Shishkin mesh, and, for the time-dependent problem, first order convergence in time in the maximum norm, again with a logarithmic factor for the Shishkin mesh. Finally, numerical examples are given to support the theoretical results.