Iterative solutions of the discretized Helmholtz equation

This research determines the efficiency with which the discretized Helmholtz equation can be solved using a preconditioned Generalized Minimum Residual (GMRES) method. The Helmholtz equation is discretized by the Green's Function Discretization (GFD) method. Both two-dimensional (2-D) and three-dimensional (3-D) Helmholtz equations are analyzed. The efficiency of the method is compared to direct solvers. It will be shown in this thesis that the preconditioned iterative method used here requires fewer operations and less computing time than direct methods for even relatively small 3-D problems. Finally, it will be shown in each of the 2-D and 3-D problems that the iterative method can solve systems of equations that are much larger than the direct methods can handle. Specifically, we demonstrate the solution of a system with more than one million equations with complex variables.