An interface penalty finite element method for elliptic interface problems on piecewise meshes

Abstract

An interface penalty finite element method (IPFEM) is proposed for elliptic interface problems, which allows to use different meshes in different subdomains separated by the interface. The transmission conditions across the interface are treated by the Nitsche’s method (or penalty technique) with some harmonic-weighted averages. Both symmetric IPFEM and non-symmetric IPFEM are analyzed. Optimal order error estimates in energy norm and L2-norm and the flux error estimate in L2-norm are proved for the symmetric IPFEM. In particular, the relative error estimate in energy norm and the flux error estimate in L2-norm are independent of the ratio of mesh sizes and the contrast of the discontinuous coefficients across the interface. Error estimates for the non-symmetric IPFEM are also obtained. Furthermore, optimal mesh sizes with respect to the error estimate in energy norm and the flux error estimate in L2-norm are discussed, respectively. Numerical examples are also provided to confirm the theoretical results and show that the IPFEM on a mesh with optimal mesh sizes may give much better approximations than the one on the quasi-uniform mesh with the same number of nodal points.