A multigrid method for the Helmholtz equation with optimized coarse grid corrections

We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let G_c denote the number of points per wavelength at the coarse level. If the coarse scale solutions are to approximate the true solutions, then the oscillatory nature of the solutions implies the requirement G_c > 2. However, in examples the requirement is more like G_c >= 10, in a trade-off involving also the amount of damping present and the number of multigrid iterations. We conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators. Standard 5-point finite differences in 2-D are our first example. A new coarse scale 9-point operator is constructed to match the fine scale phase speeds. We then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator. The required G_c is reduced from about 10 to about 3.5, with less damping present so that waves propagate over > 100 wavelengths in the new scheme. Next we consider extensions of the method to more general cases. In 3-D comparable results are obtained with standard 7-point differences and optimized 27-point coarse grid operators, leading to an order of magnitude reduction in the number of unknowns for the coarsest scale linear system. Finally we show how to include PML boundary layers, using a regular grid finite element method. Matching coarse scale operators can easily be constructed for other discretizations. The method is therefore potentially useful for a large class of discretized high-frequency Helmholtz equations.Comment: Coarse scale operators are simplified and only standard smoothers used in v3; 5 figures, 12 table

A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory

We develop a new dispersion minimizing compact finite difference scheme for the Helmholtz equation in 2 and 3 dimensions. The scheme is based on a newly developed ray theory for difference equations. A discrete Helmholtz operator and a discrete operator to be applied to the source and the wavefields are constructed. Their coefficients are piecewise polynomial functions of $hk$, chosen such that phase and amplitude errors are minimal. The phase errors of the scheme are very small, approximately as small as those of the 2-D quasi-stabilized FEM method and substantially smaller than those of alternatives in 3-D, assuming the same number of gridpoints per wavelength is used. In numerical experiments, accurate solutions are obtained in constant and smoothly varying media using meshes with only five to six points per wavelength and wave propagation over hundreds of wavelengths. When used as a coarse level discretization in a multigrid method the scheme can even be used with downto three points per wavelength. Tests on 3-D examples with up to $10^8$ degrees of freedom show that with a recently developed hybrid solver, the use of coarser meshes can lead to corresponding savings in computation time, resulting in good simulation times compared to the literature.Comment: 33 pages, 12 figures, 6 table