We consider a family of conforming space-time finite element discretizations for the wave equation based on splines of maximal regularity in time. Traditional techniques may require a CFL condition to guarantee stability. Recent works by O. Steinbach and M. Zank (2018), and S. Fraschini, G. Loli, A. Moiola, and G. Sangalli (2023), have introduced unconditionally stable schemes by adding non-consistent penalty terms to the underlying bilinear form. Stability and error analysis have been carried out for lowest order discrete spaces. While higher order methods have shown promising properties through numerical testing, their rigorous analysis was still missing. In this paper, we address this stability analysis by studying the properties of the condition number of a family of matrices associated with the time discretization. For each spline order, we derive explicit estimates of both the CFL condition required in the unstabilized case and the penalty term that minimises the consistency error in the stabilized case. Numerical tests confirm the sharpness of our results.

翻译：本文研究了一类基于时间方向最大正则性样条的波动方程时空协调有限元离散方法。传统技术可能需要CFL条件来保证稳定性。O. Steinbach与M. Zank（2018）以及S. Fraschini、G. Loli、A. Moiola和G. Sangalli（2023）的最新工作通过向基本双线性形式添加非协调惩罚项，提出了无条件稳定格式。现有稳定性与误差分析仅针对最低阶离散空间建立。虽然高阶方法在数值实验中已展现出良好特性，但其严格理论分析尚属空白。本文通过研究时间离散化相关矩阵族的条件数特性，系统解决了该稳定性分析问题。针对每个样条阶数，我们分别推导了非稳定情形下所需CFL条件的显式估计，以及稳定情形下使协调误差最小化的惩罚项显式估计。数值实验验证了所得结果的精确性。