Supercloseness in a balanced norm of the NIPG method on Shishkin mesh for a reaction diffusion problem

Abstract

For the error analysis of singularly perturbed reaction-diffusion problems, the balanced norm, which is stronger than the usual energy norm, is introduced to correctly reflect the behavior of the layer. In this paper, we study the convergence in a balanced norm for nonsymmetric interior penalty Galerkin (NIPG) method for the first time. For this purpose, a new interpolation is designed, which consists of a Gauß Lobatto interpolation in the layer, and a locally weighted L2 projection outside the layer. On that basis, by properly defining the penalty parameters at different nodes on a Shishkin mesh, we obtain the supercloseness of almost k+12 order, and prove the convergence of optimal order in a balanced norm. Here k is the degree of polynomials. Numerical experiments verify the main conclusion.