An energy preserving mixed finite element for Maxwell"s equations with nonlinear conductivity

Abstract:

An energy-preserving mixed finite element is constructed to solve the Maxwell"s equations with nonlinear conductivity. This finite element is obtained by discretizing the first order formulation of the Maxwell"s equations in space based on the finite element exterior calculus as well as the continuous time Galerkin method, which can be viewed as a modification of the Crank-Nicolson method, is used to discretize the time. Then we obtain a full discrete scheme preserving the total energy exactly when the source term vanishes. The mixed finite element method can preserve the magnetic Gauss law exactly. Based on a projection-based quasi-interpolation operator, the optimal order convergence of the method is established. Finally, numerical examples are presented to exemplify the theoretical results.