A Fast Solver for the Fractional Helmholtz Equation
Abstract
The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead of the standard Laplacian. Recently, it has been established that such a fractional Helmholtz problem better captures the underlying behavior in geophysical electromagnetics. We establish the well-posedness and regularity of this problem. We introduce a hybrid spectral-finite element approach to discretize it and show well-posedness of the discrete system. In addition, we derive a priori discretization error estimates. Finally, we introduce an efficient solver that scales as well as the best possible solver for the classical integer-order Helmholtz equation. We conclude with several illustrative examples that confirm our theoretical findings.
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Supplementary Material
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Index of Supplementary Materials
Title of paper: A Fast Solver for the Fractional Helmholtz Equation
Authors: Harbir Antil, Marta D'Elia, Christian Glusa, Bart Van Bloemen Waanders, and Chester J. Weiss
File: supp.pdf
Type: PDF
Contents: Detailed proofs of auxiliary results
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Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A1362 - A1388
DOI: 10.1137/19M1302351
ISSN (online): 1095-7197
Copyright
© 2021, Society for Industrial and Applied Mathematics.
History
Submitted: 25 February 2020
Accepted: 12 January 2021
Published online: 26 April 2021
Keywords
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Authors
Funding Information
Sandia National Laboratories https://doi.org/10.13039/100006234 : DE-NA0003525
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1818772, DMS-1913004
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