Stability of conforming space-time isogeometric methods for the wave equation

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.