Iterative solution of multiple radiation and scattering problems in structural acoustics using a block quasi-minimal residual algorithm

Finite-element discretizations of time-harmonic acoustic wave problems in exterior domains result in large sparse systems of linear equations with complex symmetric coefficient matrices. In many situations, these matrix problems need to be solved repeatedly for different right-hand sides, but with the same coefficient matrix. Recently, Freund and Malhotra have proposed a block quasi-minimal residual (BL-QMR) algorithm [13] for the iterative solution of non-Hermitian linear systems with multiple right-hand sides. The BL-QMR algorithm is a block Krylov-subspace iterative method that incorporates deflation to delete linearly and almost linearly dependent vectors in the underlying block Krylov sequences. In this paper, we describe a J -symmetric variant of the BL-QMR algorithm that introduces important simplifications for the case when the coefficient matrix is symmetric with respect to a bilinear form induced by a certain matrix J . In particular, the J -symmetric variant includes the complex symmetric form of BL-QMR as a special case. We identify suitable preconditioners for the BL-QMR algorithm applied to multiple radiation and scattering problems. Our numerical tests with the preconditioned BL-QMR algorithm for such multiple linear systems show that, instead of solving each of the linear systems individually, it is significantly more efficient to employ the block version of the iterative method. Moreover, the numerical results clearly illustrate the importance of deflation and its effect on iterative convergence.