Adaptive refinement for hp–Version Trefftz discontinuous Galerkin methods for the homogeneous Helmholtz problem

Abstract

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.

Application of Adapted-Bubbles to the Helmholtz Equation with Large Wavenumbers in 2D

“…There is a great effort in literature to overcome these difficulties. Among the discretization techniques, there are finite difference [10,24], finite element [1,18,27], discontinuous Galerkin [6,11], virtual element [21], and boundary element methods [19]. At the same time, there is a great effort to develop efficient preconditioners, such as multigrid [4,7,8,15,26] and domain decomposition methods [16,17,25].…”