Efficient finite element method for 2D singularly perturbed parabolic convection diffusion problems with discontinuous source term

Abstract

This article presents a numerical solution for a specific class of 2D parabolic singularly perturbed convection-diffusion problems with a special interior line source. The proposed approach employs the alternating direction implicit type operator splitting streamline-diffusion finite element method (SDFEM), offering a viable solution to alleviate computational complexity and high storage requirements encountered in higher-dimensional problems. The overall stability of the two-step method is established, while a piecewise-uniform Shishkin mesh is employed for spatial domain discretization. By carefully selecting the stabilization parameter, an \(\varepsilon \)-uniform error estimate is derived, accounting for the influence of the time step-interval which is essential to maintain the method’s stability. To validate the theoretical error estimate, numerical investigation are conducted, showcasing the effectiveness of the proposed method. This research contributes to advancing the understanding and numerical treatment of this specific class of 2D parabolic singularly perturbed convection-diffusion problems, shedding light on the intricate dynamics and behavior of the system in the presence of a special interior line source.