Space-time discretization of the wave equation in a second-order-in-time formulation: a conforming, unconditionally stable method

In this paper, we propose and analyze a new conforming space-time Galerkin discretization of the wave equation, which is based on a second-order-in-time variational formulation. Our method requires at least $C^1$-regularity in time, and it is shown to be unconditionally stable for all choices of discrete spaces that satisfy standard approximation properties and inverse inequalities, such as spline spaces. In particular, the variational formulation of the associated ordinary differential equation is coercive. The proposed method yields error estimates with respect to the mesh size that are suboptimal by one order in standard Sobolev norms. However, for certain choices of approximation spaces, it achieves quasi-optimal estimates. In particular, we prove this for $C^1$-regular splines of even polynomial degree, and provide numerical evidence suggesting that the same behavior holds for splines with maximal regularity, irrespective of the degree. Numerical results are provided to support the theoretical findings and demonstrate the sharpness of the estimates.