Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

de Prenter, Frits; Verhoosel, C. V. Verhoosel; van Brummelen, E. H. van Brummelen; Evans, J.A.; Messe, C. Messe; Benzaken, J. Benzaken; Maute, K.

Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

Journal article (2020)

Authors

Frits de Prenter Reden B.V., Eindhoven University of Technology

C. V. Verhoosel Verhoosel Eindhoven University of Technology

E. H. van Brummelen van Brummelen Eindhoven University of Technology

J.A. Evans Ann & H.J. Smead, University of Colorado

C. Messe Messe Deutsches Zentrum für Luft- und Raumfahrt (DLR), University of Colorado

J. Benzaken Benzaken University of Colorado, Walt Disney Animation Studios

K. Maute University of Colorado

Affiliation

External organisation

Multigrid Preconditioner Immersed finite element method Fictitious domain method Iterative solver

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Published Date

2020

Language

English

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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.