Robust and Efficient Mixed Hybrid Discontinuous Finite Element Methods for Elliptic Interface Problems

Abstract

Because of the discontinuity of the interface problems, it is natural to apply the\r discontinuous Galerkin (DG) finite element methods to solve those problems. In this work, both\r fitted and unfitted mixed hybrid discontinuous Galerkin (MHDG) finite element methods are\r proposed to solve the elliptic interface problems. For the fitted case, the problems can be solved\r directly by MHDG method. For the unfitted case, the broken basis functions (unnecessary to\r satisfy the jump conditions) are introduced to those elements which are cut across by interface,\r the weights depending on the volume fractions of cut elements and the different diffusions (or\r material heterogeneities) are used to stabilize the method, and the idea of the Nitsche's penalty\r method is applied to guarantee the jumps on the interface parts of cut elements. Unlike the\r immersed interface finite element methods (IIFEM), the two jump conditions are enforced weakly\r in our variational formulations. So, our unfitted interface MHDG method can be applied more\r easily than IIFEM to general cases, particularly when the immersed basis function cannot be\r constructed. Numerical results on convergence and sensitivities of both interface location within a\r cut element and material heterogeneities show that the proposed methods are robust and efficient\r for interface problems.