Supercloseness in a balanced norm of finite element methods on Shishkin and Bakhvalov-Shishkin rectangular meshes for reaction-diffusion problems

We present a convergence analysis for finite element methods of any order, which are applied on Shishkin mesh and Bakhvalov-Shishkin mesh to a singularly perturbed reaction-diffusion problem. A new interpolant is introduced for analysis in the balanced norm. By means of this interpolant and some superconvergence estimations, we prove supercloseness results in the cases of Shishkin mesh and Bakhvalov-Shishkin mesh. Numerical experiments also support these theoretical results.