Scalable multigrid methods for immersed finite element methods and immersed isogeometric analysis

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 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.

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Cited by in corpus (16)

• Isogeometric Analysis on V-reps: first results
• Mathematical foundations of adaptive isogeometric analysis
• Industrial scale large eddy simulations (LES) with adaptive octree meshes using immersogeometric analysis
• Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach
• A minimal stabilization procedure for Isogeometric methods on trimmed geometries
• Coupling of non-conforming trimmed isogeometric Kirchhoff-Love shells via a projected super-penalty approach
• A comparison of smooth basis constructions for isogeometric analysis
• Residual-based error estimation and adaptivity for stabilized immersed isogeometric analysis using truncated hierarchical B-splines
• Direct Immersogeometric Fluid Flow and Heat Transfer Analysis of Objects Represented by Point Clouds
• Extension Operators for Trimmed Spline Spaces
• Topology-preserving Scan-based Immersed Isogeometric Analysis
• An EigenValue Stabilization Technique for Immersed Boundary Finite Element Methods in Explicit Dynamics
• A multimesh finite element method for the Navier-Stokes equations based on projection methods
• Analysis of optimal preconditioners for CutFEM
• Finite Cell Method for functionally graded materials based on V-models and homogenized microstructures
• A ghost-point smoothing strategy for geometric multigrid on curved boundaries