Residual-Based Adaptivity and PWDG Methods for the Helmholtz Equation

Abstract

We present a study of two residual a posteriori error indicators for the Plane Wave Discontinuous Galerkin (PWDG) method for the Helmholtz equation. In particular we study the h-version of PWDG in which the number of plane wave directions per element is kept fixed. First we use a slight modification of the appropriate a priori analysis to determine a residual indicator. Numerical tests show that this is reliable but pessimistic in that the ratio between the true error and the indicator increases as the mesh is refined. We therefore introduce a new analysis based on the observation that sufficiently many plane waves can approximate piecewise linear functions as the mesh is refined. Numerical results demonstrate an improvement in the efficiency of the indicators.

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

“…Results from [24] indicate that the wave-based DGM exhibits exponential convergence with respect to the number of degrees of freedom for very general element shapes, including in the presence of strong mesh refinements. This paves the way to for the development of fully automatic hp-adaptive versions of the method and a posteriori error estimators are being investigated [25]. In this work, the wave-based DGM from [18] is used.…”

A plane wave discontinuous Galerkin method with a Dirichlet-to-Neumann boundary condition for the scattering problem in acoustics

“…for any arbitrary z h ∈ P W (T h ). To approximate the right hand side of (53) we follow the idea introduced in Lemma 5.3 of [22] and Lemma 3.10 of [13]: Let z c h be the conforming piecewise linear finite element interpolant of z ∈ H 2 (Ω). Then we can find a z h ∈ P W (T h ) that can approximate z c h .…”

“…where we have used the quasi-uniformity of the mesh and the stability estimate (23). To estimate the term A N (e N h , z c h − z h ), we use results from Lemma 5.4 of [22]. By similar arguments used in the estimation of A N (e N h , z N − z c h ), and using the second inequality of Lemma 2.5 to bound z N L ∞ (Ω) , we get…”

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

“…Noting that z ∈ H 2 /3+s (Ω), 0 < s ≤ 1 /2, cf. [16], we recall the following (second) a posteriori error bound from [21].…”

“…The purpose of this article is to develop an efficient hp-adaptive refinement algorithm for TDG methods applied to the homogeneous Helmholtz problem; we will specifically consider the ultraweak variational formulation with plane wave basis functions [8]. Within the adaptive procedure, elements will be marked for refinement based on employing an empirical a posteriori error indicator, stimulated by the upper bounds derived in [21] for the h-version of the TDG method. For the h-version of the plane wave discontinuous Galerkin method, incorporating Lagrange multipliers, a similar error indicator has been presented in [2].…”