Adaptive Mesh Refinement and Superconvergence for Two-Dimensional Interface Problems

Adaptive mesh refinement and the Borgers algorithm are combined to generate a body-fitted mesh which can resolve the interface with fine geometric details. Standard linear finite element method based on such body-fitted meshes is applied to the elliptic interface problem and proven to be superclose to the linear interpolation of the exact solution. Based on this superconvergence result, a maximal norm error estimate of order one and half is obtained without using the discrete maximum principle. The data structure and meshing algorithms, including local refinement and coarsening, are very simple. In particular, no tree structure is needed. An efficient solver for solving the resulting linear algebraic systems is also developed and shown be robust with respect to both the problem size and the jump of the diffusion coefficients.