ABSTRACT
Recent work by Sun, et al. [I J has lead to a novel method for the efficient solution of large eigenmodc problems on parallel computers, and this method has been implemented in the Omega3P solver. The method, called lSIL (Inexact Shift and Invert Lanczos), uses a standard shift-invert Lanczos process to solve the generalized eigenvalue problem, except that the solution of the shifted linear system is done iteratively only to low precision (I o-2 residual). The inexact Lanczos process will usually stagnate before the desired eigenvector residual is obtained, at which point the Jacobi Orthogonal
With an appropriate choice of linear solver, such as PCG, lSIL scales well to many processors. For large problems the computational time is dominated by the time required to solve the shifted linear system. The performance of PCG on matrices generated using the finite element method in electromagnctics is strongly dependent on the particular problem and, of course, the preconditioner. Scalability requirements limit the options for preconditioners, with some of the more effective strategies on serial machines involving factorization of the coefficient matrix being less desirable when the problem is to be solved on a parallel computer. Given these constraints, the Gauss-Seidel method is a good choice for preconditioner, with a number of steps from I to 5 depending on the situation (each step requires essentially two matrix-vector multiplies). With this linear solver, however, it can still take many thousands of iterations to reduce the initial residual by a factor of I 00. Indeed, even in some cases involving relatively small meshes, it is possible to exceed 50,000 iterations without suitable convergence.
