A Fast Solver for the Fractional Helmholtz Equation

Technical Report · 03 October 2019

DOI:https://doi.org/10.2172/1569144· OSTI ID:1569144

Antil, Harbir ^[1]; D'Elia, Marta ^[2]; Glusa, Christian Alexander ^[2]; van Bloemen Waanders, Bart G. ^[2]; Weiss, Chester Joseph ^[2]

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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 Laboratories (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
• OSTI ID: 1569144
• Report Number(s): SAND--2019-10998R; 679632
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
fractional Helmholtz equation
spectral fractional Laplacian