Multigrid Methods for Standing Wave Equations

Ira Livshits

Weizmann Institute of Science, Rehovot, Israel.


In this work we present a new multigrid solver for the standing wave equations with radiation boundary conditions. The straightforward application of standard multigrid technique cannot provide an efficient solver to these equations, since some special Fourier error components (with frequencies depending on the wave number) have no efficient reduction: They are almost invisible for any relaxation on the fine grids and have no accurate approximation on the coarse grids. Therefore, this type of error needs special treatment. Our approach is based on the fact that each such problematic error can be factorized by representing it as the product of a certain high-frequency Fourier component and a smooth envelope function. The idea is then to reduce this type of error by approximating there smooth envelope functions on the coarse grids. An additional advantage of this approach is that it allows a natural introduction of the radiation boundary conditions.