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

Abstract

This article is devoted to the study of a fully discrete A‐ ϕ finite element method to solve nonlinear Maxwell's equations based on backward Euler discretization in time and nodal finite elements in space. The nonlinearity is owing to a field‐dependent conductivity with the power‐law form | E | α − 1 , 0 < α < 1 . We design a nonlinear time‐discrete scheme for approximation in suitable function spaces. We show the well‐posedness of the problem, prove the convergence of the semidiscrete scheme based on the boundedness of the second derivative in the dual space and derive its error estimate. The Minty–Browder technique is introduced to obtain the convergence of the nonlinear term. Finally, we discuss the error estimate for the fully discretized problem and support the theoretical result by two numerical experiments. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2083–2108, 2014

Boundary data identification for an electromagnetic problem by means of the potential field method

“…Besides, it can also be changed into potential formulations by means of decomposition of the field E or H (called the A-φ or T -ψ method) and thus nodal finite elements are used to solve it numerically, cf. [1,[4][5][6][7][8][9][10]16,17]. There are several advantages for the potential method.…”