Supercloseness of Continuous Interior Penalty Methods on Shishkin Triangular Meshes and Hybrid Meshes for Singularly Perturbed Problems with Characteristic Layers

Abstract

A singularly perturbed convection–diffusion problem posed on the unit square is solved using a continuous interior penalty (CIP) method. The mesh used is a Shishkin triangular mesh or a Shishkin hybrid mesh consisting of triangles and rectangles. For the CIP method, a variant of Oswald interpolation operator is introduced for a discrete inf-sup stability, which is proved in a new norm stronger than the the usual CIP norm. This stability and a new cancellation technique enable new supercloseness results for the CIP method: the computed solutions on the triangular mesh and the hybrid mesh are shown to be 3/2 order and 2 order (up to a logarithmic factor) convergent in the new norm to the interpolants of the true solution, respectively. These convergence orders are uniformly valid with respect to the diffusion parameter and imply that for the Shishkin mesh the hybrid mesh is superior to the triangular one. Numerical experiments illustrate these theoretical results.