TY - JOUR

T1 - A fast spectral subtractional solver for elliptic equations

AU - Braverman, Elena

AU - Epstein, Boris

AU - Israeli, Moshe

AU - Averbuch, Amir

N1 - Funding Information:
The research of E.B. was partially supported by the University Research Grant of the University of Calgary. The research of M.I. was partially supported by the VPR fund for promotion of research at the Technion.

PY - 2004/8

Y1 - 2004/8

N2 - The paper presents a fast subtractional spectral algorithm for the solution of the Poisson equation and the Helmholtz equation which does not require an extension of the original domain. It takes O(N2 log N) operations, where N is the number of collocation points in each direction. The method is based on the eigenfunction expansion of the right hand side with integration and the successive solution of the corresponding homogeneous equation using Modified Fourier Method. Both the right hand side and the boundary conditions are not assumed to have any periodicity properties. This algorithm is used as a preconditioner for the iterative solution of elliptic equations with non-constant coefficients. The procedure enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points. We also apply the basic solver to the solution of the Poisson equation in complex geometries.

AB - The paper presents a fast subtractional spectral algorithm for the solution of the Poisson equation and the Helmholtz equation which does not require an extension of the original domain. It takes O(N2 log N) operations, where N is the number of collocation points in each direction. The method is based on the eigenfunction expansion of the right hand side with integration and the successive solution of the corresponding homogeneous equation using Modified Fourier Method. Both the right hand side and the boundary conditions are not assumed to have any periodicity properties. This algorithm is used as a preconditioner for the iterative solution of elliptic equations with non-constant coefficients. The procedure enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points. We also apply the basic solver to the solution of the Poisson equation in complex geometries.

KW - Equations in complex geometries

KW - Fast spectral direct solver

KW - Preconditioned iterative algorithm for elliptic equations

KW - The Poisson equation

KW - The modified helmholtz equation

UR - http://www.scopus.com/inward/record.url?scp=7544231055&partnerID=8YFLogxK

U2 - 10.1023/B:JOMP.0000027957.39059.6b

DO - 10.1023/B:JOMP.0000027957.39059.6b

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AN - SCOPUS:7544231055

SN - 0885-7474

VL - 21

SP - 91

EP - 128

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

IS - 1

ER -