A fast Helmholtz solver for high wavenumbers

To overcome computational demands to describe acoustic wave propagation accurately, one can formulate the Helmholtz equation in terms of amplitude and phase [2]. For predominantly propagating waves, the amplitude and phase are smooth functions for any wavenumber and solving the equations in terms of these variables enormously reduces calculation times. For a benchmark problem, converged solutions were shown for wavenumber times meshsize (kh-)values of upto 15000, i.e. 2500 waves per element! The coupled set of non-linear equations were difficult to solve however; the solution is not unique, convergence problems may occur and a good initial approximation is essential for the solver to converge. In this paper, we propose a solution to these problems using the natural logarithm of the amplitude and the phase as unknown variables, ensuring uniqueness of the solution. We present the new theory, a FEM discretization and show numerical results for a benchmark problem, as well as results for two more realistic problems.