Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

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

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

23 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)807-838
Number of pages32
JournalComputational Mechanics
Issue number3
Publication statusPublished - 2020


  • Fictitious domain method
  • Immersed finite element method
  • Iterative solver
  • Multigrid
  • Preconditioner

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