 
Summary: SIAM J. SCI. COMPUT. c 2005 Society for Industrial and Applied Mathematics
Vol. 26, No. 5, pp. 15041524
A HIERARCHICAL 3D DIRECT HELMHOLTZ SOLVER BY
DOMAIN DECOMPOSITION AND MODIFIED FOURIER METHOD
E. BRAVERMAN, M. ISRAELI, AND A. AVERBUCH§
Abstract. 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 3D method which was
developed in our earlier papers (see, for example, SIAM J. Sci. Comput., 20 (1999), pp. 22372260).
This method is based on the application of the discrete Fourier transform accompanied by a subtrac
tion technique. For high accuracy the subdomain boundary conditions must be compatible with the
specified inhomogeneous righthand 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.
Key words. fast threedimensional solver, Helmholtz equation, Fourier method, domain de
composition
