Low-Order Preconditioning of High-Order Triangular Finite Elements
Abstract
We propose a new formulation of a low-order elliptic preconditioner for high-order triangular elements. In the preconditioner, the nodes of the low-order finite element problem do not necessarily coincide with the high-order nodes. Instead, the two spaces are connected using least squares projection operators. The effectiveness of the preconditioner is demonstrated to be highly sensitive to the location and number of vertices used to construct the low-order finite element mesh on each high-order element. We treat the number of low-order vertices and their locations as optimizable quantities and chose them to minimize the condition number of the preconditioned stiffness matrix on the reference element. We present computational results that demonstrate that the condition number of the preconditioned high-order stiffness matrix on the reference element can be improved. The best performing preconditioners are formed with low-order finite element meshes that have more vertices than the high-order element has degrees of freedom, and have nodes grouped close to the element edges.
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Published In
SIAM Journal on Scientific Computing
Pages: A4040 - A4059
DOI: 10.1137/17M1149444
ISSN (online): 1095-7197
Copyright
© 2018, Society for Industrial and Applied Mathematics.
History
Submitted: 27 September 2017
Accepted: 31 August 2018
Published online: 4 December 2018
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U.S. Department of Energy https://doi.org/10.13039/100000015 : 17-SC-20-SC
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