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A Fast Solver for the Fractional Helmholtz Equation

Technical Report ·
DOI:https://doi.org/10.2172/1569144· OSTI ID:1569144
 [1];  [2];  [2];  [2];  [2]
  1. George Mason Univ., Fairfax, VA (United States)
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead ofthe 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 finite element-spectral 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.

Research Organization:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program; National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR)
DOE Contract Number:
AC04-94AL85000; NA0003525; DMS-1818772; DMS-1913004; FA9550-19-1-0036
OSTI ID:
1569144
Report Number(s):
SAND-2019-10998R; 679632
Country of Publication:
United States
Language:
English