Adaptive Refinement for $hp$-Version Trefftz Discontinuous Galerkin Methods for the Homogeneous Helmholtz Problem
For computational scientists solving Helmholtz problems, this work provides an adaptive method that improves efficiency, though it is incremental as it combines existing techniques.
The authors developed an hp-adaptive refinement procedure for Trefftz discontinuous Galerkin methods for the Helmholtz problem, combining h-refinement, p-refinement, and directional adaptivity. Numerical experiments demonstrated efficiency using an empirical error indicator.
In this article we develop an $hp$-adaptive refinement procedure for Trefftz discontinuous Galerkin methods applied to the homogeneous Helmholtz problem. Our approach combines not only mesh subdivision (h-refinement) and local basis enrichment (p-refinement), but also incorporates local directional adaptivity, whereby the elementwise plane wave basis is aligned with the dominant scattering direction. Numerical experiments based on employing an empirical a posteriori error indicator clearly highlight the efficiency of the proposed approach for various examples.