A Multigrid Method for Unfitted Finite Element Discretizations of Elliptic Interface Problems

Abstract

We consider discrete Poisson interface problems resulting from linear unfitted finite elements, also called cut finite elements. Three of these unfitted finite element methods, known from the literature, are studied. Two of these are suitable only for small jumps in the diffusion coefficient, and the third one has a robustness property that makes it appropriate also for interface problems with large coefficient jumps. All three methods rely on Nitsche's method to incorporate the interface conditions. The main topic of the paper is the development of a multigrid method, based on a novel prolongation operator for the unfitted finite element space and an interface smoother that is designed to yield robustness for large jumps in the diffusion coefficients. Numerical results are presented which illustrate efficiency of this multigrid method and demonstrate its robustness properties with respect to variation of the mesh size, location of the interface, and contrast in the diffusion coefficients.