Pointwise error estimates for a singularly perturbed time-dependent semilinear reaction-diffusion problem
IMA Journal of Numerical Analysis
An initial-boundary-value problem for a semilinear reaction-diffusion equation is considered. Its diffusion parameter ε 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
... estigate 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 ε 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.