Conforming space-time variational formulations for the wave equation

Matteo Ferrari (Università di Vienna)

Abstract: In this talk, we consider space-time conforming Galerkin discretizations of the acoustic wave equation. Unlike time-stepping methods, the time variable here is treated as an additional dimension.
Our goal is to design numerical schemes that are unconditionally stable, quasi-optimally convergent, and suitable for efficient implementation. Ideally, we aim for a formulation that guarantees these properties under minimal assumptions on the discrete spaces, allowing for broad applicability. In particular, we are interested in methods that go beyond piecewise continuous polynomials, extending also to spline functions of arbitrary regularity.
We compare the properties, advantages, and limitations of three continuous variational formulations, distinguished by their treatment of the temporal part:
-a second-order-in-time scheme without integration by parts in time,
-a first-order-in-time scheme,
-a second-order-in-time scheme with integration by parts in time.
We will present recent results and highlight some open problems.
This talk is based on joint works with I. Perugia and E. Zampa.