A Hierarchical 3-D Poisson Modified Fourier Solver by Domain Decomposition

We present a Domain Decomposition non-iterative solver for the Poisson equation in a 3-D rectangular box. The solution domain is divided into mostly parallelepiped subdomains. In each subdomain a particular solution of the non-homogeneous equation is first computed by a fast spectral method. 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. In this paper we present the matching algorithm for two boxes which is a basis of the domain decomposition scheme. The hierarchical approach is convenient for parallelization and minimizes the global communication. The algorithm requires O(N³ log N) operations, where N is the number of grid points in each direction.