### Abstract

^{1}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.

Language | English |
---|---|

Pages | A325-A345 |

Number of pages | 21 |

Journal | SIAM Journal on Scientific Computing |

Volume | 38 |

Issue number | 1 |

DOIs | |

Publication status | Published - 28 Jan 2016 |

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### Keywords

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

### Cite this

*SIAM Journal on Scientific Computing*,

*38*(1), A325-A345. https://doi.org/10.1137/15M1014887

}

*SIAM Journal on Scientific Computing*, vol. 38, no. 1, pp. A325-A345. https://doi.org/10.1137/15M1014887

**Efficient block preconditioning for a C1 finite element discretisation of the Dirichlet biharmonic problem.** / Pestana, J.; Muddle, R.; Heil, M.; Tisseur, F.; Mihajlovic, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Efficient block preconditioning for a C1 finite element discretisation of the Dirichlet biharmonic problem

AU - Pestana, J.

AU - Muddle, R.

AU - Heil, M.

AU - Tisseur, F.

AU - Mihajlovic, M.

PY - 2016/1/28

Y1 - 2016/1/28

N2 - 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.

AB - 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.

KW - biharmonic equation

KW - hermite bicubic finite elements

KW - block preconditioning

KW - conjugate gradient method

KW - algebraic multigrid

UR - http://epubs.siam.org/loi/sisc

U2 - 10.1137/15M1014887

DO - 10.1137/15M1014887

M3 - Article

VL - 38

SP - A325-A345

JO - SIAM Journal on Scientific Computing

T2 - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 1

ER -