Balanced Norm Convergence of DGFEM for Singularly Perturbed Parabolic Problems With Discontinuous Diffusion

ABSTRACT This paper investigates singularly perturbed parabolic reaction‐diffusion problems (SPP‐RDPs) featuring discontinuous variable diffusion, reaction, and source functions. We employ the discontinuous Galerkin finite element method (DGFEM), specifically the nonsymmetric interior penalty Galerkin method (NIPG) for the spatial discretization, and the discontinuous Galerkin (DG) method for temporal discretization. The primary contribution is the establishment of uniform convergence in the balanced norm, which provides a more accurate error representation than traditional energy norms. Notably, this is the first study of the balanced norm error estimates for DGFEM applied to SPP‐RDPs. Numerical experiments are conducted to validate the theoretical results and demonstrate the effectiveness of the proposed scheme.