Pointwise error estimates for a singularly perturbed time-dependent semilinear reaction-diffusion problem

An initial boundary-value problem for a semilinear reaction-diffusion equation is considered. Its diffusion parameter epsilon(2) is arbitrarily small, which induces initial and boundary layers. It is shown that the conventional implicit method might produce incorrect computed solutions on uniform meshes. Therefore we propose a stabilized method that yields a unique qualitatively correct solution on any mesh. Constructing discrete upper and lower solutions, we prove existence and investigate the accuracy of discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is established that the two considered methods enjoy second-order convergence in space and first-order convergence in time (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm, if epsilon < C(N-1 + M-1/2), where N and M are the numbers of mesh intervals in the space and time directions, respectively. Numerical results are presented that support the theoretical conclusions.