Efficient Implementation of the 3D Helmholtz equation in C++/PETSc

Abstract

The paper describes the comparison of different preconditioners for the solution of the Helmholtz equation with Krylov subspace methods in three dimensions. The solution of this equation has applications in microwave imaging and microwave hyperthermia for cancer detection and treatment. Due to the challenging nature of the Helmholtz equation, we employ a frequency and convergence analysis in two and three dimensions. We examine the sensitivity of the equation to various parameters and determine the effectiveness of various preconditioners. The use of finite difference approximation and preconditioned Krylov subspace methods allows for a convergence order of 2 to be achieved. The numerical results provide support for the aforementioned statement. Provided that there are no issues with resonant frequencies the desired convergence is achieved. This is applicable to the results obtained for different parameter functions and frequencies, as well as for two- and three-dimensional problems.