Approximation by dominant wave directions in plane wave methods

Plane wave methods have become an established tool for the solution of Helmholtz problems in homogeneous media. The idea is to approximate the solution in each element with a linear combination of plane waves, which are roughly equally spaced in all directions. The main advantage of plane wave methods is that they require significantly fewer degrees of freedom per unknown than standard finite elements. However, for many wave problems there are only a few dominant wave directions, which can be found using ray tracing or other high-frequency methods. Based on arguments from high-frequency asymptotics we show that dominant plane waves can be a suitable approximation basis if multiplied (modulated) by small degree polynomials. We explore this approach for two example problems and compare its performance against standard equispaced plane wave basis sets. Finally, we present an example with smoothly varying speed of sound, which demonstrates that for such problems approximations based on polynomially modulated dominant wave directions can far outperform standard plane wave methods, which are not well suited for handling problems in varying media