Convergence analysis of a hp-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary

Serge Nicaise (1) , Jérôme Tomezyk (2)

1 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015
2 UVHC - Université de Valenciennes et du Hainaut-Cambrésis

We consider a non conforming hp-finite element approximation of a variational formulation of the time-harmonic Maxwell equations with impedance boundary conditions proposed in [5, §4.5.d]. The advantages of this formulation is that the variational space is embedded in H 1 as soon as the boundary is smooth enough (in particular it holds for domains with an analytic boundary) and standard shift theorem from [8] can be applied since the associated boundary value problem is elliptic. Finally in order to perform a wavenumber explicit error analysis of our problem, a splitting lemma and an estimation of the adjoint approximation quantity are proved by adapting to our system the results from [16, 17] obtained for the Helmholtz equation. Some numerical tests that illustrate our theoretical results are also presented. Analytic regularity results with bounds explicit in the wavenumber of the solution of a general elliptic system with lower order terms depending on the wavenumber are need and hence proved.

Mots clés: Maxwell equations; absorbing boundary conditions; smooth domains; finite elements

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Serge Nicaise, Jérôme Tomezyk. Convergence analysis of a hp-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary. 2019. ⟨hal-02063271⟩