Abstract
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 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.
Highlights
Immersed methods are useful tools to avoid laborious and computationally expensive procedures for the generation of body-fitted finite element discretizations or analysis-suitable NURBS geometries in isogeometric analysis, for problems on complex, moving, or implicitly defined geometries
This contribution develops a geometric multigrid preconditioner that enables iterative solutions for higher-order immersed finite element methods at a computational cost that is linear with the number of degrees of freedom
This is an improvement with respect to state-of-the-art preconditioning techniques for immersed finite element methods and immersed isogeometric analysis, as these are either restricted to linear discretizations [47,48,49] or provide convergence rates that are dependent on the grid size [35,50,51]
Summary
Immersed methods are useful tools to avoid laborious and computationally expensive procedures for the generation of body-fitted finite element discretizations or analysis-suitable NURBS geometries in isogeometric analysis, for problems on complex, moving, or implicitly defined geometries. The main objective of this contribution is to develop a geometric multigrid preconditioning technique that is applicable to higher-order immersed finite element methods with conventional, isogeometric, and locally refined basis functions This preconditioner enables iterative solution methods with a convergence rate that is unaffected by either the cut elements or the grid size, such that the solution is obtained at a computational cost that scales linearly with the number of degrees of freedom (DOFs).
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