A Multigrid Method for a Nitsche-based Extended Finite Element Method

Abstract

We present a tailored multigrid method for linear problems stemming from a Nitsche-based extended finite element method (XFEM). Our multigrid method is robust with respect to highly varying coefficients and the number of interfaces in a domain. It shows level independent convergence rates when applied to different variants of Nitsche’s method. Generally, multigrid methods require a hierarchy of finite element (FE) spaces which can be created geometrically using a hierarchy of nested meshes. However, in the XFEM framework, standard multigrid methods might demonstrate poor convergence properties if the hierarchy of FE spaces employed is not nested. We design a prolongation operator for the multigrid method in such a way that it can accommodate the discontinuities across the interfaces in the XFEM framework and recursively induces a nested FE space hierarchy. The prolongation operator is constructed using so-called pseudo-L^2-projections; as common, the adjoint of the prolongation operator is employed as the restriction operator.