Dr Carolina Urzua Torres will talk about 'Paving the way to a T-coercive method for the wave equation'

Space-time Galerkin methods are gradually becoming popular, since they allow adaptivity and parallelization in space and time simultaneously. A lot of progress has been made for parabolic problems, and its success has motivated an increased interest in finding space-time formulations for the wave equation that lead to unconditionally stable discretizations. In this talk I will discuss some of the challenges that arise and some recent work in this direction.

In particular, I will present what we see as a first step toward introducing a space-time transformation operator $T$ that establishes $T$-coercivity for the weak variational formulation of the wave equation in space and time on bounded Lipschitz domains. As a model problem, we study the ordinary differential equation (ODE) $u'' + \mu u = f$ for $\mu>0$, which is linked to the wave equation via a Fourier expansion in space. For its weak formulation, we introduce a transformation operator $T_\mu$ that establishes $T_\mu$-coercivity of the bilinear form yielding an unconditionally stable Galerkin-Bubnov formulation with error estimates independent of $\mu$. The novelty of the current approach is the explicit dependence of the transformation on $\mu$ which, when extended to the framework of partial differential equations, yields an operator acting in both time and space. We pay particular attention to keeping the trial space as a standard Sobolev space, simplifying the error analysis, while only the test space is modified.
The theoretical results are complemented by numerical examples.