TY - JOUR

T1 - A hierarchical 3-D direct Helmholtz solver by domain decomposition and modified Fourier method

AU - Braverman, E.

AU - Israeli, M.

AU - Averbuch, A.

PY - 2005

Y1 - 2005

N2 - The paper contains a noniterative solver for the Helmholtz and the modified Helmholtz equations in a hexahedron. The solver is based on domain decomposition. The solution domain is divided into mostly parallelepiped subdomains. In each subdomain a particular solution of the nonhomogeneous Helmholtz equation is first computed by a fast spectral 3-D method which was developed in our earlier papers (see, for example, SIAM J. Sci. Comput., 20 (1999), pp. 2237-2260). This method is based on the application of the discrete Fourier transform accompanied by a subtraction technique. For high accuracy the subdomain boundary conditions must be compatible with the specified inhomogeneous right-hand side at the edges of all the interfaces. In the following steps the partial solutions are hierarchically matched. At each step pairs of adjacent subdomains are merged into larger units. The paper describes in detail the matching algorithm for two boxes which is a basis for the domain decomposition scheme. The hierarchical approach is convenient for parallelization and can minimize the global communication. The algorithm requires O(N3 log N) operations, where N is the number of grid points in each direction.

AB - The paper contains a noniterative solver for the Helmholtz and the modified Helmholtz equations in a hexahedron. The solver is based on domain decomposition. The solution domain is divided into mostly parallelepiped subdomains. In each subdomain a particular solution of the nonhomogeneous Helmholtz equation is first computed by a fast spectral 3-D method which was developed in our earlier papers (see, for example, SIAM J. Sci. Comput., 20 (1999), pp. 2237-2260). This method is based on the application of the discrete Fourier transform accompanied by a subtraction technique. For high accuracy the subdomain boundary conditions must be compatible with the specified inhomogeneous right-hand side at the edges of all the interfaces. In the following steps the partial solutions are hierarchically matched. At each step pairs of adjacent subdomains are merged into larger units. The paper describes in detail the matching algorithm for two boxes which is a basis for the domain decomposition scheme. The hierarchical approach is convenient for parallelization and can minimize the global communication. The algorithm requires O(N3 log N) operations, where N is the number of grid points in each direction.

KW - Domain decomposition

KW - Fast three-dimensional solver

KW - Fourier method

KW - Helmholtz equation

UR - http://www.scopus.com/inward/record.url?scp=25444483323&partnerID=8YFLogxK

U2 - 10.1137/S1064827502417039

DO - 10.1137/S1064827502417039

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AN - SCOPUS:25444483323

SN - 1064-8275

VL - 26

SP - 1504

EP - 1524

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

IS - 5

ER -