On the approximation of electromagnetic fields by edge finite elements. Part 1: sharp interpolation results for low-regularity fields

Patrick Ciarlet (1)

1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation

We propose sharp results on the numerical approximation of low-regularity electromagnetic fields by edge finite elements. We consider general geometrical settings, including topologically non-trivial domains or domains with a non-connected boundary. In the model, the electric permittivity and magnetic per-meability are symmetric, tensor-valued, piecewise smooth coefficients. In all cases, the error can be bounded by h δ times a constant, where h is the mesh-size, for some exponent δ ∈]0, 1] that depends both on the geometry and on the coefficients. It relies either on classical estimates when δ > 1/2, or on a new combined interpolation operator when δ < 1/2. The optimality of the value of δ is discussed with respect to abstract shift theorems. In some simple configurations , typically for scalar-valued permittivity and permeability, the value of δ can be further characterized. This paper is the first one in a series dealing with the approximation of electromagnetic fields by edge finite elements.

Mots clés: Maxwell's equations; interface problem; edge elements; interpolation operators; error estimates

Domaines: Analyse numérique [math.NA]

Patrick Ciarlet. On the approximation of electromagnetic fields by edge finite elements. Part 1: sharp interpolation results for low-regularity fields. 2015. ⟨hal-01176476v3⟩