A hierarchical preconditioner for wave problems in quasilinear complexity
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by
Boris Bonev, Jan S. Hesthaven
2021
Abstract
The paper introduces a novel, hierarchical preconditioner based on nested
dissection and hierarchical matrix compression. The preconditioner is intended
for continuous and discontinuous Galerkin formulations of elliptic problems. We
exploit the property that Schur complements arising in such problems can be
well approximated by hierarchical matrices. An approximate factorization can be
computed matrix-free and in a (quasi-)linear number of operations. The nested
dissection is specifically designed to aid the factorization process using
hierarchical matrices. We demonstrate the viability of the preconditioner on a
range of 2D problems, including the Helmholtz equation and the elastic wave
equation. Throughout all tests, including wave phenomena with high wavenumbers,
the generalized minimal residual method (GMRES) with the proposed
preconditioner converges in a very low number of iterations. We demonstrate
that this is due to the hierarchical nature of our approach which makes the
high wavenumber limit manageable.
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