Abstract
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(N 3:log:N) operations, where N is the number of grid points in each direction.
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REFERENCES
Averbuch, A., Braverman, E., and Israeli, M. (2000). Parallel adaptive solution of a Poisson equation with multiwavelets. SIAM J. Sci. Comput 22, 1053–1086.
Averbuch, A., Braverman, E., and Israeli, M. (2001). A new low communication parallel algorithm for elliptic partial differential equations. In Parallel Computational Fluid Dynamics. Trends and Applications, Proceedings of the Parallel CFD 2000 Conference, Trondheim, Norway, May 22–25, Elsevier, Amsterdam, pp. 275–282.
Averbuch, A., Israeli, M., and Vozovoi, L. (1997). On fast direct elliptic solver by modified Fourier method. Numerical Algorithms 15, 287–313.
Averbuch, A., Israeli, M., and Vozovoi, L. (1998). A fast Poisson solver of arbitrary order accuracy in rectangular regions. SIAM J. Sci. Comput. 19, 933–952.
Braverman, E., Israeli, M., Averbuch, A., and Vozovoi, L. (1998). A fast 3-D Poisson solver of arbitrary order accuracy. J. Comput. Phys. 144, 109–136.
Braverman, E., Israeli, M., and Averbuch, A. (1999). A fast spectral solver for 3-D Helmholtz equation. SIAM J. Sci. Comput. 20, 2237–2260.
Gottlieb, D., and Orszag, S. A. (1977). Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia.
Greengard, L., and Lee, J.-Y. (1996). A direct adaptive Poisson solver of arbitrary order accuracy. J. Comput. Phys. 125, 415–424.
Greengard, L., and Rokhlin, V. (1987). A fast algorithm for particle simulations. J. Comput. Phys. 73, 325–348.
Karniadakis, E. G., Israeli, M., and Orszag, S. A. (1991). High-Order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97, 414–443.
McKenney, A., Greengard, L., and Mayo, A. (1995). A fast Poisson solver for complex geometries. J. Comput. Phys. 118, 348–355.
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Israeli, M., Braverman, E. & Averbuch, A. A Hierarchical 3-D Poisson Modified Fourier Solver by Domain Decomposition. Journal of Scientific Computing 17, 471–479 (2002). https://doi.org/10.1023/A:1015102109496
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DOI: https://doi.org/10.1023/A:1015102109496