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We study the convergence of multigrid schemes for the Helmholtz equation,focusing in particular on the choice of the coarse scale operators. Let G_cdenote the number of points per wavelength at the coarse level. If the coarsescale solutions are to approximate the true solutions, then the oscillatorynature of the solutions implies the requirement G_c > 2. However, in examplesthe requirement is more like G_c >= 10, in a trade-off involving also theamount of damping present and the number of multigrid iterations. We conjecturethat this is caused by the difference in phase speeds between the coarse andfine scale operators. Standard 5-point finite differences in 2-D are our firstexample. A new coarse scale 9-point operator is constructed to match the finescale phase speeds. We then compare phase speeds and multigrid performance ofstandard schemes with a scheme using the new operator. The required G_c isreduced from about 10 to about 3.5, with less damping present so that wavespropagate over > 100 wavelengths in the new scheme. Next we consider extensionsof the method to more general cases. In 3-D comparable results are obtainedwith standard 7-point differences and optimized 27-point coarse grid operators,leading to an order of magnitude reduction in the number of unknowns for thecoarsest scale linear system. Finally we show how to include PML boundarylayers, using a regular grid finite element method. Matching coarse scaleoperators can easily be constructed for other discretizations. The method istherefore potentially useful for a large class of discretized high-frequencyHelmholtz equations.

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