A novel finite element scheme is studied for solving the time-dependent Maxwell's equations on unstructured grids efficiently. Similar to the traditional Yee scheme, the method has one degree of freedom for most edges and a sparse inverse mass matrix. This allows for an efficient realization by explicit time-stepping without solving linear systems. The method is constructed by algebraic reduction of another underlying finite element scheme which involves two degrees of freedom for every edge. Mass-lumping and additional modifications are used in the construction of this method to allow for the mentioned algebraic reduction in the presence of source terms and lossy media later on. A full error analysis of the underlying method is developed which by construction also carries over to the reduced scheme and allows to prove convergence rates for the latter. The efficiency and accuracy of both methods are illustrated by numerical tests. The proposed schemes and their analysis can be extended to structured grids and in special cases the reduced method turns out to be algebraically equivalent to the Yee scheme. The analysis of this paper highlights possible difficulties in extensions of the Yee scheme to non-orthogonal or unstructured grids, discontinuous material parameters, and non-smooth source terms, and also offers potential remedies.

翻译：本文研究了一种新型有限元方案，用于高效求解非结构网格上的时变麦克斯韦方程组。与传统Yee格式类似，该方法对大多数棱边仅赋予一个自由度，并具有稀疏逆质量矩阵。这使得通过显式时间步进实现高效求解成为可能，无需解线性方程组。该方案通过对另一种每个棱边包含两个自由度的底层有限元方案进行代数约简而构建。在构建过程中，采用了质量集中与额外修正，使得后续处理源项和有耗介质时能实现上述代数约简。本文对底层方法进行了完整的误差分析，该分析自然适用于约简方案，并可证明后者的收敛率。数值实验验证了两种方法的效率与精度。所提出的方案及其分析可推广至结构网格，且在特殊情形下，约简方法在代数上等价于Yee格式。本文的分析揭示了将Yee格式推广至非正交或非结构网格、非连续材料参数以及非光滑源项时可能面临的困难，并提出了潜在解决方法。