Evaluation of Iterative-Solutions used for Three-Dimensional Volume Integral Equation

Abstract

The impedance matrix generated by the discretization of volume integral equation is usually nonsymmetrical and dense. When the direct matrix solver such as Gaussian elimination is employed to solve the impedance matrix, O(N²) memory and O(N³) operations are required, where N is the number of unknowns. Therefore, the direct matrix solver is not suitable for the practical solution of large-scale problems by the volume integral equation. The implement of the iterative-methods is the realistic solution to improve the calculation efficiency in solving the problems. In this paper, we evaluate six well-known iterative-methods in solving the matrix equation obtained by the discretization of volume integral equation. We investigate the convergence characteristics of the residual-norms in this evaluation. In the methods based on Lanczos process, it is found that the convergence characteristics of the residual-norms become unstable under some conditions. In the methods based on Arnoldi process, it is found that the convergence characteristics of the residual-norms are always stable under various conditions. Since we found that GMRES is the most effective iterative-method in solving the matrix equation obtained by the discretization of volume integral equation, we particularly investigate the convergence characteristic of GMRES based on Arnoldi process.