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：我们考虑一类基于时间方向上最大正则样条的波动方程适定时空有限元离散格式。传统方法通常需要满足CFL条件以保证稳定性。O. Steinbach与M. Zank（2018年）以及S. Fraschini、G. Loli、A. Moiola、G. Sangalli（2023年）等学者近期通过向底层双线性形式添加非一致性罚项，提出了无条件稳定格式。目前仅针对最低阶离散空间完成了稳定性和误差分析。尽管高阶方法通过数值试验展现出优良性质，但其严格分析仍然缺失。本文通过研究时间离散关联矩阵族的条件数性质来解决这一稳定性分析问题。针对每个样条阶次，我们给出了非稳定化情形下所需CFL条件的显式估计，以及稳定化情形下使相容性误差最小化的罚项表达式。数值试验验证了结果的精确性。