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：本文针对麦克斯韦方程组的初边值问题，提出了一种高阶样条几何方法。该方法在结构保持意义上离散了连续系统表述中涉及的两个de Rham微分形式序列，体现了几何特性。安培-麦克斯韦方程和法拉第方程均需严格成立，同时利用两个离散Hodge星算子确保系统可解。通过选取合适的样条空间并借助外微分概念，本文引入了一种非标准显式时间格式：其基于具有Kronecker积结构的矩阵的线性系统求解，而非像标准文献中那样依赖于质量矩阵。这些矩阵源自离散设定中外积（楔积）的应用，且无论区域几何或材料参数如何，均呈现Kronecker积结构。所提出的方案保持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，系统研究了新方案的计算优势。