Efficient block preconditioning for a C1 finite element discretisation of the Dirichlet biharmonic problem

J. Pestana, R. Muddle, M. Heil, F. Tisseur, M. Mihajlovic

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

We present an efficient block preconditioner for the two-dimensional biharmonic Dirichlet problem discretised by C1 bicubic Hermite finite elements. In this formulation each node in the mesh has four different degrees of freedom (DOFs). Grouping DOFs of the same type together leads to a natural blocking of the Galerkin coefficient matrix. Based on this block structure, we develop two preconditioners: a 2×2 block diagonal preconditioner (BD) and a block bordered diagonal (BBD) preconditioner. We prove mesh-independent bounds for the spectra of the BD-preconditioned Galerkin matrix under certain conditions. The eigenvalue analysis is based on the fact that the proposed preconditioner, like the coefficient matrix itself, is symmetric positive definite and is assembled from element matrices. We demonstrate the effectiveness of an inexact version of the BBD preconditioner, which exhibits near-optimal scaling in terms of computational cost with respect to the discrete problem size. Finally, we study robustness of this preconditioner with respect to element stretching, domain distortion and non-convex domains.
LanguageEnglish
PagesA325-A345
Number of pages21
JournalSIAM Journal on Scientific Computing
Volume38
Issue number1
DOIs
Publication statusPublished - 28 Jan 2016

Fingerprint

Biharmonic Problem
Finite Element Discretization
Preconditioning
Preconditioner
Dirichlet Problem
Stretching
Costs
Degree of freedom
Mesh
Optimal Scaling
Eigenvalue Analysis
Block Structure
Coefficient
Hermite
Galerkin
Grouping
Positive definite
Computational Cost
Finite Element
Robustness

Keywords

  • biharmonic equation
  • hermite bicubic finite elements
  • block preconditioning
  • conjugate gradient method
  • algebraic multigrid

Cite this

Pestana, J. ; Muddle, R. ; Heil, M. ; Tisseur, F. ; Mihajlovic, M. / Efficient block preconditioning for a C1 finite element discretisation of the Dirichlet biharmonic problem. In: SIAM Journal on Scientific Computing. 2016 ; Vol. 38, No. 1. pp. A325-A345.
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Efficient block preconditioning for a C1 finite element discretisation of the Dirichlet biharmonic problem. / Pestana, J.; Muddle, R.; Heil, M.; Tisseur, F.; Mihajlovic, M.

In: SIAM Journal on Scientific Computing, Vol. 38, No. 1, 28.01.2016, p. A325-A345.

Research output: Contribution to journalArticle

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