A condensed generalized finite element method (CGFEM) for interface problems

Extensive developments on various generalizations of the Finite Element Method (FEM) for the interface problems, with unfitted mesh, have been made in the last few decades. Typical generalizations and techniques include the immersed FEM, the penalized FEM, the generalized or the extended FEM (GFEM/XFEM). The GFEM/XFEM achieves good approximation by augmenting the finite element approximation space with enrichment functions, which require additional degrees of freedom (DOF). However these additional DOFs, in general, deteriorate the conditioning of the GFEM/XFEM. In this study, we propose a condensed GFEM (CGFEM) for the interface problem based on the partition of unity method and a local least square scheme. Compared to the conventional GFEM/XFEM, the CGFEM possesses the same number of DOFs as in the FEM, yields good approximation, and is well-conditioned. In comparison with the immersed FEM, the construction of shape functions of CGFEM is independent of the equation types and the material coefficients of the interface problem. As a result, the CGFEM could be applied to more general interface problems, e.g., problems with anisotropic materials and elasticity systems, in a unified approach. Moreover, the shape functions of CGFEM are continuous, and thus do not need any penalty terms and parameters. The optimal order of convergence of the CGFEM has been analyzed and established theoretically in this paper. Numerical experiments for isotropic and anisotropic interface problems and comparisons with the conventional GFEM/XFEM have also been presented to illuminate the theory and effectiveness of CGFEM.