Numerical Treatment of Coupled Singularly Perturbed Reaction–Diffusion Problems Using DDG Method on Shishkin Mesh

In this paper, we investigate the application of the direct discontinuous Galerkin (DDG) method for the numerical approximation of singularly perturbed coupled reaction–diffusion systems. These problems exhibit overlapping and interacting boundary layers, which pose significant challenges for numerical approximation. To effectively capture these layers, we employ DDG method on a piecewise-uniform Shishkin mesh. We prove that the proposed numerical scheme is uniformly convergent with order [Formula: see text] in the [Formula: see text]-weighted energy norm, where [Formula: see text] denotes the degree of the piecewise polynomial approximation. Numerical experiments are provided to illustrate and support the theoretical results.