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

Abstract

In this paper, we study a numerical scheme to solve coupled Maxwell's equations with a nonlinear conductivity. This model plays an important role in the study of type‐II superconductors. The approximation scheme is based on backward Euler discretization in time and mixed conforming finite elements in space. We will prove convergence of this scheme to the unique weak solution of the problem and develop the corresponding error estimates. As a next step, we study the stability of the scheme in the quasi‐static limit ϵ → 0 and present the corresponding convergence rate. Finally, we support the theory by several numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.

Fully discrete A‐ϕ finite element method for Maxwell's equations with nonlinear conductivity

“…The authors proved its convergence based on the boundedness of the second derivative in the dual space by the Minty–Browder technique. Some similar nonlinear problems have been studied relevant to quasistatic Maxwell's equations, resulting in a nonlinear parabolic partial differential equation (see, e.g., ). In , quasistatic Maxwell's equations were discussed in the context of a power‐law nonlinearity with 0 < α < 1.…”

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

“…A power law 2430 JIANGXING WANG for the conductivity have been used to modeling the type-II superconductors [17] and modelling the charge-density wave state of NbSe3 [9]. The well poseness of the system (1a)-(1b) have been investigated in [5][6][7]26]. The existence of the weak solution for a nonlinear function J(E) = σ(|E|)E, with σ(s) monotonically increasing is shown in [7,26].…”

“…The existence of the weak solution for a nonlinear function J(E) = σ(|E|)E, with σ(s) monotonically increasing is shown in [7,26]. In [6], the authors presented the existence and uniqueness of the system (1a)-(1b) when J(E) = σ(|E|)E, with σ(s) monotonically increasing. In [5], the authors proved the existence of the weak solution for J(E) = σ(|E|)E, with σ(s) = |s| α .…”

“…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].…”

“…[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]. A fully-discrete (T, ψ) − ψ e finite element decoupled scheme was developed to solve time-dependent eddy current problem with multipleconnected conductors [2].…”