NUMERICAL MICROLOCAL ANALYSIS BY FAST GAUSSIAN WAVE PACKET TRANSFORMS AND APPLICATION TO HIGH-FREQUENCY HELMHOLTZ PROBLEMS

摘要

We develop a novel numerical microlocal analysis (NMLA) method using fast Gaussian wave packet transforms. Our new NMLA method extracts ray directions at discrete locations by analyzing highly oscillatory wavefields with fast Gaussian wave packet transforms. Theoretically, to achieve the same accuracy, the novel NMLA has the same time complexity as the original NMLA, which is based on local plane-wave analysis for Helmholtz problems, but the new NMLA is applicable to generic oscillatory wavefields and is straightforward to implement. We apply the new NMLA to high-frequency Helmholtz problems, so that we are able to extract directions from a reduced-frequency solution and further incorporate these directions into a ray-based interior-penalty discontinuous Galerkin method to solve for high-frequency solutions. In this way, we observe no apparent pollution effects. We provide numerical results in 2 dimensions to justify the claims.