Preconditioned Krylov and Gauss-Seidel solutions of response matrix equations

The use of preconditioned Krylov methods is examined as an alternative to the partitioned matrix acceleration applied to red-black Gauss Seidel (RBGS) iteration that is presently used in the variational nodal code, VARIANT. We employ the GMRES algorithm to treat non-symmetric response matrix equations. A pre conditioner is formulated for the within-group diffusion equation which is equivalent to partitioned matrix acceleration of RBGS iterations. We employ the pre conditioner, which closely parallels two-level p multigrid, to improve RBGS and GMRES algorithms. Of the accelerated algorithms, GMRES converges with less computational effort than RBGS and therefore is chosen for further development. The p multigrid pre conditioner requires response matrices with two or more degrees of freedom (DOF) per interface that are polynomials, which are both orthogonal and hierarchical. It is therefore not directly applicable to very fine mesh calculations that are both slow to converge and that are often modeled with response matrices with only one DOF per interface. Orthogonal matrix aggregation (OMA) is introduced to circumvent this difficulty by combining N×N fine mesh response matrices with one DOF per interface into a coarse mesh response matrix with N orthogonal DOF per interface. Numerical results show that OMA used alone or in combination with p multigrid preconditioning substantially accelerates GMRES solutions. (author)