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Zbl 1089.78023
Carpentieri, B.; Duff, I.S.; Giraud, L.; Sylvand, G.
Combining fast multipole techniques and an approximate inverse preconditioner for large electromagnetism calculations. (English)
[J] SIAM J. Sci. Comput. 27, No. 3, 774-792 (2005). ISSN 1064-8275; ISSN 1095-7197/e

Summary: The boundary element method has become a popular tool for the solution of Maxwell's equations in electromagnetism. From a linear algebra point of view, this leads to the solution of large dense complex linear systems, where the unknowns are associated with the edges of the mesh defined on the surface of the illuminated object. In this paper, we address the iterative solution of these linear systems via preconditioned Krylov solvers. Our primary focus is on the design of an efficient parallelizable preconditioner. In that respect, we consider an approximate inverse method based on the Frobenius-norm minimization. The preconditioner is constructed from a sparse approximation of the dense coefficient matrix, and the patterns both for the preconditioner and for the coefficient matrix are computed a priori using geometric information from the mesh. We describe how such a preconditioner can be naturally implemented in a parallel code that implements the multipole technique for the matrix-vector product calculation. We investigate the numerical scalability of our preconditioner on realistic industrial test problems and show that it exhibits some limitations on very large problems of size close to one million unknowns. To improve its robustness on those large problems we propose an embedded iterative scheme that combines nested GMRES solvers with different fast multipole computations. We show through extensive numerical experiments that this new scheme is extremely robust at affordable memory and CPU costs for the solution of very large and challenging problems.

MSC 2000:: *78M15 Boundary element methods (optics)
65F10 Iterative methods for linear systems
65R20 Integral equations (numerical methods)
65N38 Boundary element methods (BVP of PDE)
78A55 Technical appl. of optics and electromagnetic theory
65Y05 Parallel computation (numerical methods)

Keywords: sparse approximate inverse preconditioner; large dense complex systems; Maxwell equations; boundary elements; fast multipole technique; frequency domain; electric field integral equations; parallel computing; large calculations

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