BDF Schemes in Stable Generalized Finite Element Methods for Parabolic Interface Problems with Moving Interfaces

There are several difficulties in generalized/extended finite element methods (GFEM/XFEM) for moving interface problems. First, the GFEM/XFEM may be unstable in a sense that condition numbers of system matrices could be much bigger than those of standard FEM. Second, they may not be robust in that the condition numbers increase rapidly as interface curves approach edges of meshes. Furthermore, time stepping schemes need carrying out carefully since both enrichment functions and enriched nodes in the GFEM/XFEM vary in time. This paper is devoted to proposing the stable and robust GFEM/XFEM with effi- cient time stepping schemes for the parabolic interface problems with moving interfaces. A so-called stable GFEM (SGFEM) developed for elliptical interface problems is extended to the parabolic interface problems for spatial discretiza- tions; while backward difference formulae (BDF) are used for the time stepping. Numerical studies demonstrate that the SGFEM with the first and second order BDF (also known as backward Euler method and BDF2) is stable, robust, and achieves optimal convergence rates. Comparisons of the proposed SGFEM with various commonly -used GFEM/XFEM are made, which show advantages of the SGFEM over the other GFEM/XFEM in aspects of stability, robustness, and convergence.