Fully discrete finite element scheme for Maxwell's equations with non-linear boundary condition

Abstract

We study a full Maxwell's system accompanied with a non-linear degenerate boundary condition, which represents a generalization of the classical Silver-Müller condition for a non-perfect conductor. The relationship between the normal components of electric E and magnetic H field obeys the following power law ν × H = ν × (|E × ν| α−1 E × ν) for some α ∈ (0, 1]. We establish the existence and uniqueness of a weak solution in a suitable function spaces under the minimal regularity assumptions on the boundary Γ and the initial data E 0 and H 0 . We design a non-linear time discrete approximation scheme and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time discretization. As a next step we study the fully discrete problem using curl-conforming edge elements and derive the corresponding error estimates. Finally we present some numerical experiments.

Convergence of the mixed finite element method for Maxwell's equations with nonlinear conductivity

“…A fully discrete finite element scheme for nonlinear Maxwell's equations is studied in , where the equations are decoupled to obtain one single PDE. In , the authors applied the mixed finite element method to study Maxwell's equations with a nonlinear boundary condition. Study of the stability of Maxwell's equations under the singular limit ϵ → 0 was first performed in and applied in to prove existence of a quasi‐static system with increasing conductivity.…”

Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations

“…Recently, finite element methods have been applied to solve nonlinear Maxwell equation (cf. [5,6,15,18,19]). Numerical schemes with backward Euler discretization in time and mixed conforming finite elements in space were discussed for nonlinear conductivity problems in [5,6].…”

Maxwell's Equations as a Scattering Passive Linear System

“…For more details see Eller, Lagnese, and Nicaise [9], Lafitte [16], or Levy [20]. The paper by Slodicka and Durand [29] considers well-posedness and numerical methods for nonlinear generalizations of the Silver-Müller boundary condition.…”