In this paper the vector finite element time-domain (VFETD) method is derived, analyzed, and validated. The VFETD method uses edge vector finite elements as a basis for the electric field and face vector finite elements as a basis for the magnetic flux density. The Galerkin method is used to convert Maxwell's equations to a coupled system of ordinary differential equations. The leapfrog method is used to advance the fields in time. The method is shown to be stable and to conserve energy and charge for arbitrary hexahedral grids. A numerical dispersion analysis shows the method to be second order accurate on distorted hexahedral grids. Several computational experiments are performed to determine the accuracy and efficiency of the method.

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In this paper, we motivate the use of high-order integration methods for finite element solutions of the time dependent Maxwell equations. In particular, we present a symplectic algorithm for the integration of the coupled first-order Maxwell equations for computing the time dependent electric and magnetic fields. Symplectic methods have the benefit of conserving total electromagnetic field energy and are, therefore, preferred over dissipative methods (such as traditional Runge-Kutta) in applications that require high-accuracy and energy conservation over long periods of time integration. We show that in the context of symplectic methods, several popular schemes can be elegantly cast in a single algorithm. We conclude with some numerical examples which demonstrate the superior performance of high-order time integration methods.Index Terms-Finite element methods, high-order methods, Maxwell equations, symplectic methods, time domain analysis.

We present a mixed vector finite element method for solving the time dependent coupled Ampere and Faraday laws of MaxwellÕs equations on unstructured hexahedral grids that employs high order discretization in both space and time. The method is of arbitrary order accuracy in space and up to 4th order accurate in time, making it well suited for electrically large problems where grid anisotropy and numerical dispersion have plagued other methods. In addition, the method correctly models both the jump discontinuities and the divergence-free properties of the electric and magnetic fields, is charge and energy conserving, conditionally stable, and free of spurious modes. Several computational experiments are performed to demonstrate the accuracy, efficiency and benefits of the method.