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.
Original language | English |
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Pages (from-to) | A325-A345 |
Number of pages | 21 |
Journal | SIAM Journal on Scientific Computing |
Volume | 38 |
Issue number | 1 |
DOIs | |
Publication status | Published - 28 Jan 2016 |
Keywords
- biharmonic equation
- hermite bicubic finite elements
- block preconditioning
- conjugate gradient method
- algebraic multigrid