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

International audienceWe 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

Wellposedness and regularity for linear Maxwell equations with surface current

We study linear time-dependent Maxwell equations on a cuboid consisting of two homogeneous subcuboids. At the interface, we allow for nonzero surface charge density and surface current. This model is a first step towards a detailed mathematical analysis of the interactionof single-layer materials with electromagnetic fields. The main results of this paper provide several wellposedness and regularity statements for the solutions of the Maxwell system. To prove the statements, we employ extension arguments using interpolation theory, as well as semigroup theory and regularity theory for elliptic transmission problems

On the approximation of electromagnetic fields by edge finite elements. Part 2: A heterogeneous ultiscale method for Maxwell’s equations

In the second part of this series of papers we consider highly oscillatory media. In this situation, the need for a triangulation that resolves all microscopic details of the medium makes standard edge finite elements impractical because of the resulting tremendous computational load. On the other hand, undersampling by using a coarse mesh might lead to inaccurate results. To overcome these diffculties and to improve the ratio between accuracy and computational costs, homogenization techniques can be used. In this paper we recall analytical homogenization results and propose a novel numerical homogenization scheme for Maxwell\u27s equations in frequency domain. This scheme follows the design principles of heterogeneous multiscale methods. We prove convergence to the effective solution of the multiscale Maxwell\u27s equations in a periodic setting and give numerical experiments in accordance to the stated results

On the Approximation of Electromagnetic Fields by Edge Finite Elements. Part 2: A Heterogeneous Multiscale Method for Maxwell's equations

International audienceIn the second part of this series of papers we consider highly oscillatory media. In this situation, the need for a triangulation that resolves all microscopic details of the medium makes standard edge finite elements impractical because of the resulting tremendous computational load. On the other hand, undersampling by using a coarse mesh might lead to inaccurate results. To overcome these difficulties and to improve the ratio between accuracy and computational costs, homogenization techniques can be used. In this paper we recall analytical homogenization results and propose a novel numerical homogenization scheme for Maxwell’s equations in frequency domain. This scheme follows the design principles of heterogeneous multiscale methods. We prove convergence to the effective solution of the multiscale Maxwell’s equations in a periodic setting and give numerical experiments in accordance to the stated results

Mathematical and numerical analyses for the div-curl and div-curlcurl problems with a sign-changing coefficient

We study the numerical approximation by edge finite elements of fields whose divergence and curl, or divergence and curl-curl, are prescribed in a bounded set $\Omega$ of $\mathbb{R}^3$, together with a boundary condition. Special attention is paid to solutions with low-regularity, in terms of the Sobolev scale $({\mathbf H}^{s}(\Omega))_{s>0}$. Among others, we consider an electromagnetic-like model including an interface between a classical medium and a metamaterial. In this setting the electric permittivity, and possibly the magnetic permeability, exhibit a sign-change at the interface. With the help of T-coercivity, we address the case of a model with one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal error estimates are derived. Thanks to these results, we are also able to analyze the classical time-harmonic Maxwell equations, with one sign-changing coefficient