skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: 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 Lab. (SNL-NM), Albuquerque, NM (United States); Sandia National Lab. (SNL-CA), Livermore, CA (United States)
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; DMS-1818772; DMS-1913004; FA9550-19-1-0036; NA0003525
OSTI ID:
1765762
Report Number(s):
SAND-2021-0383J; 693329
Journal Information:
SIAM Journal on Scientific Computing, Vol. 43, Issue 2; ISSN 1064-8275
Publisher:
SIAMCopyright Statement
Country of Publication:
United States
Language:
English

References (17)

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

Similar Records

A Fast Solver for the Fractional Helmholtz Equation
Technical Report · Thu Oct 03 00:00:00 EDT 2019 · OSTI ID:1765762
+2 more
Bilevel optimization, deep learning and fractional Laplacian regularization with applications in tomography
Journal Article · Wed Apr 29 00:00:00 EDT 2020 · Inverse Problems · OSTI ID:1765762
Risk-Averse Control of Fractional Diffusion with Uncertain Exponent
Journal Article · Mon Mar 29 00:00:00 EDT 2021 · SIAM Journal on Control and Optimization · OSTI ID:1765762

Related Subjects

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