Andrea Moiola: A space-time continuous and coercive formulation for the wave equation

Abstract

We propose a new space-time variational formulation for wave equation initial-boundary value problems. The key property is that the formulation is coercive (sign-definite) and continuous in a norm stronger than H1(Q), Q being the space-time cylinder. Coercivity holds for constant-coefficient impedance cavity problems posed in star-shaped domains, and for a class of impedance-Dirichlet problems. The formulation is defined using simple Morawetz multipliers and its coercivity is proved with elementary analytical tools, following earlier work on the Helmholtz equation. The formulation can be stably discretised with any H2(Q)-conforming discrete space, leading to quasi-optimal space-time Galerkin schemes. Several numerical experiments show the excellent properties of the method. We also present a continuous-interior-penalty variant, a posteriori error estimators, and adaptive discretisations.

This is a joint work with Paolo Bignardi and Theophile Chaumont-Frelet.

[1] Bignardi, Moiola, A space-time continuous and coercive formulation for the wave equation, Numerische Mathematik 2025. https://doi.org/10.1007/s00211-025-01478-3

[2] Bignardi, Space-time Morawetz formulations for the wave equation, PhD thesis, University of Pavia 2025. https://iris.unipv.it/handle/11571/1519237