The Influence of Nitsche Stabilization on Geometric Multigrid for the Finite Cell Method

TL;DR

This paper investigates how the stabilization parameter in Nitsche's method influences the performance of a geometric multigrid solver for the finite cell method, highlighting the importance of parameter choice for solver efficiency.

Contribution

It introduces a local eigenvalue-based estimate for the stabilization parameter, improving multigrid convergence in immersed finite element methods.

Findings

Local estimate leads to rapid convergence

Stability parameter significantly affects solver performance

Robust local estimate improves multilevel solver efficiency

Abstract

Immersed finite element methods have been developed as a means to circumvent the costly mesh generation required in conventional finite element analysis. However, the numerical ill-conditioning of the resultant linear system of equations in such methods poses a challenge for iterative solvers. In this work, we focus on the finite cell method (FCM) with adaptive quadrature, adaptive mesh refinement (AMR) and Nitsche's method for the weak imposition of boundary conditions. An adaptive geometric multigrid solver is employed for the discretized problem. We study the influence of the mesh-dependent stabilization parameter in Nitsche's method on the performance of the geometric multigrid solver and its implications for the multilevel setup in general. A global and a local estimate based on generalized eigenvalue problems are used to choose the stabilization parameter. We find that the convergence rate of the solver is significantly affected by the stabilization parameter, the choice of the estimate and how the stabilization parameter is handled in multilevel configurations. The local estimate, computed on each grid, is found to be a robust method and leads to rapid convergence of the geometric multigrid solver.