Robust and highly scalable parallel solution of the Helmholtz equation with large wave numbers

Numerical solution of the Helmholtz equation is a challenging computational task, particularly when the wave number is large. For two-dimensional problems, direct methods provide satisfactory solutions, but large three-dimensional problems become unmanageable. In this work, the block-parallel CARP-CG algorithm [Parallel Computing 36, 2010] is applied to the Helmholtz equation with large wave numbers. The effectiveness of this algorithm is shown both theoretically and practically, with numerical experiments on two- and three-dimensional domains, including heterogeneous cases, and a wide range of wave numbers. A second-order finite difference discretization scheme is used, leading to a complex, nonsymmetric and indefinite linear system. CARP-CG is both robust and efficient on the tested problems. On a fixed grid, its scalability improves as the wave number increases. Additionally, when the number of grid points per wavelength is fixed, the number of iterations increases linearly with the wave number. Convergence rates for heterogeneous cases are similar to those of homogeneous cases. CARP-CG also outperforms, at all wave numbers, one of the leading methods, based on the shifted Laplacian preconditioner with a complex shift and solved with a multigrid.