WG‐FEM With ADI‐Type Splitting: A High‐Order Approach for Two‐Parameter Singularly Perturbed 2D Parabolic Problems on Bakhvalov Mesh

ABSTRACT This study presents a novel computational approach that integrates the operator splitting alternate direction implicit (ADI) scheme with a high‐order weak Galerkin finite element method to solve a two‐dimensional singularly perturbed parabolic reaction‐convection‐diffusion problem involving two small parameters. The ADI technique decomposes the original 2D problem into a sequence of 1D subproblems along each spatial direction, which are then discretized using the weak Galerkin finite element method on a nonuniform Bakhvalov‐type mesh. For time discretization, the Crank‐Nicolson scheme is applied separately in each spatial direction alongside the weak Galerkin method. Theoretical error analysis, based on a local projection, ensures spatial accuracy of order and second‐order temporal convergence under the balanced norm. Numerical experiments validate the efficiency of the proposed method and confirm the theoretical error estimates.