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

• Published in: International journal for numerical methods in fluids Vol. 95; no. 7; pp. 1035 - 1053
• Main Authors: Saberi, S., Meschke, G., Vogel, A.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 01.07.2023
• Subjects: Differential equations; Differential geometry; domain decomposition; finite cell method; Finite element method; Fluid flow; geometric multigrid; Mathematical models; Methods; Multigrid methods; Navier-Stokes equations; Partial differential equations; Robustness (mathematics); saddle‐point problems; unfitted finite element method
• ISSN: 0271-2091; 1097-0363; 1097-0363
• DOI: 10.1002/fld.5180

The 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. Adaptive geometric multigrid methods are employed for the solution of the mixed finite cell formulation of the Stokes and Navier–Stokes equations. We propose a cell‐based and a cutcell‐based Vanka‐type smoother variant and discuss their performance in the multigrid cycle for linear and nonlinear benchmarks. The solver is robust in terms of convergence and iteration count independent of problem size.