High-order extended finite element methods for solving interface problems

In this paper, we study arbitrary order extended finite element (XFE) methods based on two discontinuous Galerkin (DG) schemes in order to solve elliptic interface problems in two and three dimensions. Optimal error estimates in the piecewise H1-norm and L2-norm are rigorously proved for both schemes. In particular, we have devised a new parameter-friendly DG-XFEM method, which means that no “sufficiently large” parameters are needed to ensure the optimal convergence of the scheme. To prove the stability of bilinear forms, we derive non-standard trace and inverse inequalities for high-order polynomials on curved sub-elements divided by the interface. This paper is adapted from the work originally post on arXiv.com by the same authors (arXiv:1604.06171). New ingredients are an optimal multigrid solver for the generated linear system and its analysis. This multigrid method converges uniformly with respect to the mesh size, and is independent of the location of the interface relative to the meshes, just like all the other estimates in this paper. Numerical examples are given to support the theoretical results.