An iterative scheme for dense, complex-symmetric, linear systems in acoustics boundary-element computations

Abstract

We consider the indirect, variational, boundary-integral formulation for the Helmholtz equation in acoustics applications. This formulation leads after discretization to a dense, complex-symmetric system of linear equations, which are typically solved by direct elimination methods (using the Bunch-Kaufman pivoting strategy for matrices in the symmetric packed-storage format, as implemented in the LAPACK library [Anderson et al., LAPACK Users' Guide, SIAM, Philadelphia, 1995]). In this paper, we describe an iterative solution procedure, which uses an adaptation of the orthogonal factorization method (Gill and Murray [in Numerical Methods for Constrained Optimization, Academic Press, New York, 1974, pp. 29-65]) in conjunction with the QMR iteration (Freund [SIAM J. Sci. Statist. Comput., 13 (1992), pp. 425-448]). This iterative procedure, which achieves rapid convergence by exploiting the structure of the matrix generated by the discretization, is competitive with the LAPACK direct elimination approach for small but difficult industrial applications on which it has been tested so far, and for larger problems the iterative procedure is expected to provide significant advantages in terms of robustness, computational cost, and parallelizability.