We introduce a high-order spline geometric approach for the initial boundary value problem for Maxwell's equations. The method is geometric in the sense that it discretizes in structure preserving fashion the two de Rham sequences of differential forms involved in the formulation of the continuous system. Both the Ampere--Maxwell and the Faraday equations are required to hold strongly, while to make the system solvable two discrete Hodge star operators are used. By exploiting the properties of the chosen spline spaces and concepts from exterior calculus, a non-standard explicit in time formulation is introduced, based on the solution of linear systems with matrices presenting Kronecker product structure, rather than mass matrices as in the standard literature. These matrices arise from the application of the exterior (wedge) product in the discrete setting, and they present Kronecker product structure independently of the geometry of the domain or the material parameters. The resulting scheme preserves the desirable energy conservation properties of the known approaches. The computational advantages of the newly proposed scheme are studied both through a complexity analysis and through numerical experiments in three dimensions.

翻译：针对麦克斯韦方程组的初边值问题，本文提出一种高阶样条几何方法。该方法在几何意义上是保持结构的离散化，它保结构地离散了连续系统表述中涉及的两个de Rham微分形式序列。安培-麦克斯韦方程和法拉第方程均被强要求满足，为使系统可解，引入了两种离散Hodge星算子。通过利用所选样条空间的性质以及外微分概念，本文提出了一种非标准的显式时间离散格式，其基础是求解具有Kronecker积结构的矩阵线性系统，而非标准文献中的质量矩阵。这些矩阵源于离散设置中外积（楔积）的应用，且其Kronecker积结构与域几何或材料参数无关。所得格式保留了已知方法中理想的能量守恒特性。通过复杂度分析和三维数值实验，本文研究了新方案的计算优势。