Accuracy Properties of the Wave‐Ray Multigrid Algorithm for Helmholtz Equations

Helmholtz equations with their highly oscillatory solutions play an important role in physics and engineering. These equations present the main computational difficulties typical to acoustic, electromagnetic, and other wave problems. They are often accompanied by radiation boundary conditions and are considered on infinite domains. Solving them numerically using standard procedures, including multigrid, is too expensive. The wave-ray multigrid algorithm efficiently solves the Helmholtz equations and naturally incorporates the radiation boundary conditions. Important accuracy properties of the wave-ray solver are discussed in this paper. Using various mode analyses, we show that, with the right choice of parameters, this algorithm can obtain an approximation to the differential solution with accuracy that equals the accuracy of the target grid discretization. Moreover, the boundary conditions can be introduced with any desired accuracy. Our theoretical conclusions are confirmed by numerical experiments.