A robust weak Galerkin finite element method for two parameter singularly perturbed parabolic problems on nonuniform meshes

Abstract

In this study, we proposed a weak Galerkin finite element method (WG-FEM) for solving two-parameter singularly perturbed parabolic problems (TP-SPPPs) of convection–diffusion–reaction type on nonuniform mesh. The WG-FEM approach incorporates an interpolation operator IN and achieves optimal order convergence in an energy-like norm for continuous and fully discrete schemes. In the fully-discrete analysis, we combined WG-FEM for spatial space on a Shishkin-type mesh with the Crank–Nicolson scheme for temporal discretization on an equidistant mesh. Since the energy norm is insufficient to capture the behavior of the boundary layer functions accurately, we have derived the optimal order of convergence in a strong balanced norm. This norm exhibits an order of O(N−1lnN)k in spatial convergence and second-order convergence in time when using an L2-projection QN as an intermediate operator. We conducted numerical examples to validate the proposed method’s theoretical findings. The results of these experiments confirmed the theoretical conclusions and demonstrated the robustness of the proposed method.