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ADDITIVE SCHWARZ METHODS FOR SERENDIPITY ELEMENTS
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DOI：10.1137/21M1415674.png)
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Abstract
En 中文
While solving PDEs with finite element methods, serendipity elements allow us to obtain the same order of accuracy as rectangular tensor-product elements with many fewer degrees of freedom. To realize the possible computational savings, we develop some additive Schwarz methods based on solving local patch problems. Adapting arguments from Pavarino for the tensor-product case, we prove that patch smoothers give conditioning estimates independent of the polynomial degree for a model problem. We also combine this with a low-order global operator to give an optimal two-grid method, with conditioning estimates independent of the mesh size and polynomial degree. The theory holds for serendipity elements in two and three dimensions and can be extended to multigrid algorithms. Numerical experiments using Firedrake and PETSc confirm this theory and demonstrate efficiency relative to standard elements.
Keywords:
serendipity elements
finite element methods
additive Schwarz methods
Firedrake
multigrid methods
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