Scalable multigrid methods for immersed finite element methods and immersed isogeometric analysis
This work addresses the ill-conditioning problem in immersed finite element methods, which is a key bottleneck for iterative solvers in computational mechanics.
The authors developed a geometric multigrid preconditioner for immersed finite element methods that achieves mesh-independent and cut-element-independent convergence rates, enabling linear computational cost scaling with degrees of freedom. The method is demonstrated for higher-order discretizations including Lagrange and isogeometric B-splines.
Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems in parallel, at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.