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. The stabilization parameter in Nitsche's method plays an important role in imposing interface conditions and also affects the condition number of the linear systems. We discuss the requirements on the stabilization parameter to ensure coercivity and review selected strategies from the literature which are used to implicitly estimate the stabilization parameter. Eventually, we compare the impact of different variations of Nitsche's method on discretization errors and condition number of the linear systems. We demonstrate the robustness of our multigrid method with respect to varying coefficients and the number of interfaces and compare it with other preconditioners.