A solver for the Helmholtz equation.
The Generalized Minimization of Residuals (GMRES) method is a powerful method for solving linear systems. Nevertheless it may have some restrictions and disabilities that arise from the complex nature of the systems we are dealing with. The 2-D Helmholtz equation with a spatially dependent wave number is one of these systems. Solving such a system may be very slow (when feasible). In this project, we seek to find the best methods for choosing preconditioners for solving the heterogeneous Helmholtz equation using GMRES (with varying preconditioner). We test a wide range of options for realistic grids for the wave number, for a range of grid sizes, using a robust and generic algorithm.
The equation we are dealing with looks as follow:
is a linear partial differential equation, which depicts waves and diffusion.
We are using several patterns for the wave-number dependency.
We use Green's function in order to find the solution for in-homogeneous Helmholtz equation. The Green's function (denoted as G), is the solution for the following equation:
For the case of Helmholtz equation (L), Green's function looks as follow:
For a given FGMRES iteraion, our algorithm conducts the following sequence:
where m_0 is the current wave number, K is the kernel we are using in order to create the Green's function and q is the right-hand-side (RHS) of the linear system Au=q.