An Unfitted $hp$-Interface Penalty Finite Element Method for Elliptic Interface Problems

Abstract

An $hp$ version of interface penalty finite element method ($hp$-IPFEM) is proposed to solve the elliptic interface problems in two and three dimensions on unfitted meshes. Error estimates in broken $H^1$ norm, which are optimal with respect to $h$ and suboptimal with respect to $p$ by half an order of $p$, are derived. Both symmetric and non-symmetric IPFEM are considered. Error estimates in $L^2$ norm are proved by the duality argument. All the estimates are independent of the location of the interface relative to the meshes. Numerical examples are provided to illustrate the performance of the method. This paper is adapted from the work originally post on arXiv.com by the same authors (arXiv:1007.2893v1).