We present a fundamental improvement of a high polynomial degree time domain cell method recently introduced by the last three authors. The published work introduced a method featuring block-diagonal system matrices where the block size and conditioning scaled poorly with respect to polynomial degree. The issue is herein bypassed by the construction of new basis functions exploiting quadrature rule based mass lumping techniques for arbitrary polynomial degrees in two dimensions for the Maxwell equations and the acoustic wave equation in the first order velocity pressure formulation. We characterize the degrees of freedom of all new discrete approximation spaces we employ for differential forms and show that the resulting block diagonal (inverse) mass matrices have block sizes independent of the polynomial degree. We demonstrate on an extensive number of examples how the new technique is applicable and efficient for large scale computations.

翻译：我们提出了对最近由最后三位作者引入的高多项式阶时域单元方法的基本改进。已发表的工作引入了一种方法，其系统矩阵为块对角形式，但块大小和条件数随多项式阶数的增加而恶化。本文通过构造新的基函数克服了这一问题，该基函数利用了基于求积规则的质量集总技术，适用于二维麦克斯韦方程组和声波方程（一阶速度-压力公式）。我们刻画了用于微分形式的离散逼近空间的所有新自由度，并证明了所得块对角（逆）质量矩阵的块大小与多项式阶数无关。通过大量实例，我们展示了新技术在大规模计算中的适用性和高效性。