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Efficient Preconditioning of hp-FEM Matrices by Hierarchical Low-Rank Approximations

Abstract

We introduce a preconditioner based on a hierarchical low-rank compression scheme of Schur complements. The construction is inspired by standard nested dissection, and relies on the assumption that the Schur complements can be approximated, to high precision, by Hierarchically-Semi-Separable matrices. We build the preconditioner as an approximate \(LDM^t\) factorization of a given matrix A, and no knowledge of A in assembled form is required by the construction. The \(LDM^t\) factorization is amenable to fast inversion, and the action of the inverse can be determined fast as well. We investigate the behavior of the preconditioner in the context of DG finite element approximations of elliptic and hyperbolic problems, with respect to both the mesh size and the order of approximation.

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Notes

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    Strictly speaking, this is a tree with missing route, i.e., a forest. In fact, if we were to introduce a single top-level box holding the entirety of the dof’s, under a purely algebraic approach, no boundary dof’s could be identified.

  2. 2.

    In fact, this is the compression threshold of each off-diagonal block of A.

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Correspondence to P. Gatto.

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Gatto, P., Hesthaven, J.S. Efficient Preconditioning of hp-FEM Matrices by Hierarchical Low-Rank Approximations. J Sci Comput 72, 49–80 (2017). https://doi.org/10.1007/s10915-016-0347-x

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Keywords

  • Preconditioned GMRES
  • Interpolative decomposition
  • Indefinite operators