The finite element method applied to the exterior Helmholtz problem on the IBM SP-1
We have implemented a parallel iterative solver using the QMR-method for solving the complex, symmetric systems of linear equations arising from the FEM-formulation of the external Holmholtz-problem. Currently the system is operational on the IBM SP-1 and workstation clusters using PVM.
In addition to the solver we have created a simple, yet powerful, framework for monitoring and debugging the iteration process on the fly. This facility has already been of great value in the early stages of the development.
In the long run we expect to enhance the FEM-implementation in a number of ways, for instance by adding a geometrically correct treatment of the boundaries. We also plan to experiment with the message passing interface (MPI) implementation now available for SP-1 machines. All in all, we believe that the system described here gives us a solid foundation for further work in this field.
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- John K. OusterHout, Tcl and the Tk-Toolkit, Draft of a book by Addison-Wesley, University of California Berkeley, 1993.Google Scholar
- Claes Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Studentlitteratur, 1987.Google Scholar
- Al Geist et al., PVM 3 User's Guide and Reference Manual, Oak Ridge National Laboratory, Oak Ridge Tennessee, May 1993.Google Scholar
- Richard Barrett et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 1993Google Scholar
- C. I. Goldstein, The Finite Element Method with Non-Uniform Mesh Sizes Applied to the Exterior Helmholtz Problem, Numer. Math, 38, 61–82, 1981.Google Scholar
- Roland W. Freund and Noël Nachtigal, QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numer. Math. 60, 315–339, 1991Google Scholar
- Roland W. Freund and Noël Nachtigal, An Implementation of the QMR Method Based on Coupled Two-Term Recurrences, SIAM J. Sci. Comput., Vol. 15, No. 2, pp. 313–337, March 1994Google Scholar