Skip to main content
OSTI.GOVU.S. Department of Energy Office of Scientific and Technical Information
U.S. Department of Energy
Office of Scientific and Technical Information
 

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:
USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR); USDOE Laboratory Directed Research and Development (LDRD) Program
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

References (18)

Regularity of spectral fractional Dirichlet and Neumann problems: Regularity of spectral problems journal October 2015
On the spectrum of two different fractional operators journal July 2014
An Extension Problem Related to the Fractional Laplacian journal August 2007
Fractional operators with inhomogeneous boundary conditions: analysis, control, and discretization journal January 2018
Regularity of Radial Extremal Solutions for Some Non-Local Semilinear Equations journal August 2011
Sobolev Spaces with Non-Muckenhoupt Weights, Fractional Elliptic Operators, and Applications journal January 2019
Extension Problem and Harnack's Inequality for Some Fractional Operators journal October 2010
Optimization with Respect to Order in a Fractional Diffusion Model: Analysis, Approximation and Algorithmic Aspects journal March 2018
Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions journal April 2010
A New Type of Identification Problems: Optimizing the Fractional Order in a Nonlocal Evolution Equation journal January 2017
Numerical approximation of fractional powers of elliptic operators journal March 2015
Preasymptotic Error Analysis of CIP-FEM and FEM for Helmholtz Equation with High Wave Number. Part II: $hp$ Version journal January 2013
Finite element interpolation of nonsmooth functions satisfying boundary conditions journal May 1990
Finite Element Methods in Local Active Control of Sound journal January 2004
Hybrid Finite Element--Spectral Method for the Fractional Laplacian: Approximation Theory and Efficient Solver journal January 2018
Symmetric Stable Processes as Traces of Degenerate Diffusion Processes journal January 1969
A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis journal August 2014
Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation journal January 2011

Similar Records

Related Subjects

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