Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

Abstract

Abstract. The Ultra Weak Variational Formulation (UWVF) of the Helmholtz equation provides avariational framework suitable for discretization using plane wave solutions of an appropriate adjoint equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz equation. In this paper we exploit the fact that the UWVF is essentially an upwind discontinuous Galerkin method to prove convergence of the solution in the special case where there is no absorbing medium present. We also provide some other estimates in the case when absorption is present, and give some simple numerical results to test the estimates. We expect that similar techniques can be used to prove error estimates for the UWVF applied to Maxwell's equations and elasticity.Mathematics Subject Classification. 65N15, 65N30, 35J05.

Plane wave discontinuous Galerkin methods: Analysis of theh-version

“…This agrees with usual statement of the UWVF in terms of unknown functions on F h , see [7], Formula 19, [10], Formula (1.4), and [22], Formula 10. Matching (2.8), (2.9), and (2.10), (2.11), we see that the original UWVF by Cessenat and Després [10] is recovered by choosing α = 1/2, β = 1/2, γ = 0, δ= 1/2.…”

“…Hence, the asymptotics considered in the present paper and in [7,10] may not be the relevant. Nevertheless, we believe that investigation of h-version convergence is an essential first step in understanding the more interesting p-version of plane wave Galerkin methods.…”

“…Here, we have followed a slightly different approach and cast the UWVF within the general class of DG methods presented in [8]. A similar perspective was adopted in [7], Section 2.…”

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

“…All these methods are of Trefftz type, namely, they are based on approximation spaces made of functions which are (locally) solutions to the considered PDE. We concentrate, in particular, on the UWVF, which has recently seen rapid algorithmic development and extensions; see [15,23,24,29,[34][35][36], and we would like to analyze its application to the time-harmonic Maxwell equations, considering general Trefftz approximation spaces.…”

“…Since the UWVF can be regarded as a discontinuous Galerkin (DG) method with Trefftz basis functions (see [15,22,24]), we briefly review some literature on standard (i.e., polynomial-based) DG methods for the time-harmonic Maxwell equations. Some of them are based on the primal curl-curl formulation of the problem, neglecting the divergence-free condition.…”