Stability and error analysis of a fully-discrete numerical method for system of 2D singularly perturbed parabolic PDEs

Abstract

This article deals with a fully-discrete numerical method for the system of 2D singularly perturbed parabolic convection-diffusion problems. The method is based on the combination of the implicit Backward-Euler scheme for the temporal derivative and the streamline-diffusion finite element method (SDFEM) for the spatial derivatives. Due to the existence of exponential boundary layers in the solution of this problem, a piecewise-uniform layer-adapted mesh is used here for the triangulation of the spatial domain. The stability of the present method has been discussed based on an appropriate stabilization parameter. The convergency of the fully-discrete SDFEM on a discrete L2(0,T;L2)-norm using the piecewise continuous linear finite element space is analyzed in a new theoretical framework. Finally, the justification of the theoretical results is done by some numerical experiments.