Inf-Sup Stable Space-Time Discretization of the Wave Equation Based on a First-Order-In-Time Variational Formulation

In this paper, we present a conforming space-time discretization of the wave equation based on a first-order-in-time variational formulation. Our method extends the scheme of French and Peterson (1996), incorporating exponential weights in time, which yield an inf-sup stability condition for arbitrary choices of discrete subspaces, including spline spaces, without restrictions on the mesh size or time step. Moreover, using elliptic projections, we derive optimal convergence rates in both the energy and $L^2$ norms for sufficiently smooth solutions and for any choice of space-time tensor product subspaces satisfying standard approximation assumptions. Numerical examples are provided to support the theoretical findings.