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A Fast Solver for the Fractional Helmholtz Equation

Journal Article · · SIAM Journal on Scientific Computing
DOI:https://doi.org/10.1137/19M1302351· OSTI ID:1765762
 [1];  [2];  [3];  [1];  [4]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research
  2. George Mason Univ., Fairfax, VA (United States)
  3. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
  4. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
+ Show Author Affiliations
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. In this work, 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 aswell as the best possible solver for the classical integer-order Helmholtz equation. We conclude withseveral illustrative examples that confirm our theoretical findings.
Research Organization:
Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Sandia National Laboratories, Livermore, CA
Sponsoring Organization:
National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR); USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
AC04-94AL85000; NA0003525
OSTI ID:
1765762
Report Number(s):
SAND--2021-0383J; 693329
Journal Information:
SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 2 Vol. 43; ISSN 1064-8275
Publisher:
SIAMCopyright Statement
Country of Publication:
United States
Language:
English

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Related Subjects

97 MATHEMATICS AND COMPUTING
Fractional Helmholtz equation
error analysis
spectral fractional Laplacian