A scalable and robust vertex-star relaxation for high-order FEM

Pavarino proved that the additive Schwarz method with vertex patches and a low-order coarse space gives a \(p\)-robust solver for symmetric and coercive problems. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introdu...

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Bibliographic Details
Published inarXiv.org
Main Authors Brubeck, Pablo D, Farrell, Patrick E
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 11.10.2021
Subjects
Cholesky factorization
Coercivity
Degrees of freedom
Finite difference method
Mathematical analysis
Polynomials
Robustness
Solvers
Stiffness matrix
Tensors
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Summary:Pavarino proved that the additive Schwarz method with vertex patches and a low-order coarse space gives a \(p\)-robust solver for symmetric and coercive problems. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the stiffness matrix for the internal degrees of freedom of each individual cell. As a result, the non-zero structure of the cell matrices is that of the graph connecting internal degrees of freedom to their projection onto the facets. In the new basis, the patch problem is as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertex-patch problems, even for very high polynomial degree. In the non-separable case, the method can be applied as a preconditioner by approximating the problem with a separable surrogate.
ISSN:2331-8422