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

E. Braverman, M. Israeli, A. Averbuch

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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.

Original languageEnglish
Pages (from-to)1504-1524
Number of pages21
JournalSIAM Journal on Scientific Computing
Issue number5
StatePublished - 2005
Externally publishedYes


  • Domain decomposition
  • Fast three-dimensional solver
  • Fourier method
  • Helmholtz equation


Dive into the research topics of 'A hierarchical 3-D direct Helmholtz solver by domain decomposition and modified Fourier method'. Together they form a unique fingerprint.

Cite this