Communications in Mathematical Sciences

Volume 15 (2017)

Number 4

Localized sparsifying preconditioner for periodic indefinite systems

Pages: 1155 – 1169

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n4.a12

Authors

Fei Liu (Institute for Computational and Mathematical Engineering, Stanford University, Stanford, Calif., U.S.A.)

Lexing Ying (Department of Mathematics and Institute for Computational and Mathematical Engineering, Stanford University, Stanford, Calif., U.S.A.)

Abstract

This paper introduces the localized sparsifying preconditioner for the pseudospectral approximations of indefinite systems on periodic structures. The work is built on top of the recently proposed sparsifying preconditioner with two major modifications. First, the local potential information is utilized to improve the accuracy of the preconditioner. Second, an FFT-based method to compute the local stencil is proposed to reduce the setup time of the algorithm. Numerical results show that the iteration number of this improved method grows only mildly as the problem size grows, which implies that solving pseudospectral approximation systems is computationally as efficient as solving sparse systems, up to a mildly growing factor.

Keywords

Helmholtz equation, high frequency waves, Schrödinger equation, periodic structure, pseudospectral approximation

2010 Mathematics Subject Classification

65F08, 65F50, 65N22

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