Supercloseness of the NIPG method on a Bakhvalov-type mesh for a singularly perturbed problem with two small parameters

Abstract

In this paper, the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh is proposed for a singularly perturbed problem with two small parameters. In order to reflect the behavior of layers more accurately, a balanced norm, rather than the common energy norm, is introduced. By selecting special penalty parameters at different mesh points, we establish the supercloseness of $k+\frac{1}{2}$ order, and prove an optimal order of uniform convergence in a balanced norm. Numerical experiments are proposed to confirm our theoretical results.