Adaptive geometric multigrid for the mixed finite cell formulation of Stokes and Navier-Stokes equations

TL;DR

This paper develops an adaptive geometric multigrid solver for the mixed finite cell formulation, effectively addressing the numerical challenges in solving Stokes and Navier-Stokes equations with unfitted finite element methods.

Contribution

It introduces two smoothers for cutcells and demonstrates the multigrid method's robustness and scalability for saddle-point problems in fluid dynamics.

Findings

Solver is independent of problem size.

Method is robust to grid hierarchy depth.

Effective smoothing of cutcells demonstrated.

Abstract

Unfitted finite element methods have emerged as a popular alternative to classical finite element methods for the solution of partial differential equations and allow modeling arbitrary geometries without the need for a boundary-conforming mesh. On the other hand, the efficient solution of the resultant system is a challenging task because of the numerical ill-conditioning that typically entails from the formulation of such methods. We use an adaptive geometric multigrid solver for the solution of the mixed finite cell formulation of saddle-point problems and investigate its convergence in the context of the Stokes and Navier-Stokes equations. We present two smoothers for the treatment of cutcells in the finite cell method and analyze their effectiveness for the model problems using a numerical benchmark. Results indicate that the presented multigrid method is capable of solving the model problems independently of the problem size and is robust with respect to the depth of the grid hierarchy.