Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions

Alexandre Ern (1, 2) , Jean-Luc Guermond (3)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
3 Texas A&M University [College Station]

We derive H(curl)-error estimates and improved L 2-error estimates for the Maxwell equations. These estimates only invoke the expected regularity pickup of the exact solution in the scale of the Sobolev spaces, which is typically lower than 1 2 and can be arbitrarily close to 0 when the material properties are heterogeneous. The key tools for the analysis are commuting quasi-interpolation operators in H(curl)-and H(div)-conforming finite element spaces and, most crucially, newly-devised quasi-interpolation operators delivering optimal estimates on the decay rate of the best-approximation error for functions with Sobolev regularity index arbitrarily close to 0. The proposed analysis entirely bypasses the technique known in the literature as the discrete compactness argument.

Mots clés: Quasi-interpolation; Aubin–Nitsche duality argument; Maxwell equations; Dis- crete Poincaré inequality; Edge finite elements; Heterogeneous coefficients

Domaines: Analyse numérique [math.NA]

Alexandre Ern, Jean-Luc Guermond. Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions. Computers & Mathematics with Applications, 2018, 75 (3), pp.918-932. ⟨10.1016/j.camwa.2017.10.017⟩. ⟨hal-01531940v2⟩