具有非线性传导率的麦克斯韦方程的一个保能量混合有限元

本文针对具有非线性传导率的麦克斯韦方程构造了一个保能量的混合有限元. 其中，对麦克斯韦方程的一阶形式, 本文直接使用有限元外微分去离散空间变量, 得到保能量的半离散格式，进而通过一个二阶连续时间Galerkin方法 (CTG) 去离散半离散格式的时间变量，得到保能量的全离散格式. 本文中的半离散和全离散格式能够精确地保持磁场的严格无散条件，具有最优收敛阶. 数值算例验证了理论结果.

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

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.