Inf-sup stable space-time discretization of the wave equation based on a first-order-in-time variational formulation

TL;DR

This paper introduces a stable and convergent space-time discretization method for the wave equation using a first-order-in-time variational formulation with exponential weights, applicable to arbitrary mesh sizes.

Contribution

It develops a conforming discretization approach that guarantees stability and quasi-optimal convergence without mesh restrictions, supported by theoretical analysis and numerical validation.

Findings

Method is unconditionally stable.

Achieves quasi-optimal convergence.

Numerical examples confirm theoretical results.

Abstract

In this paper, we present a conforming space-time discretization of the wave equation based on a first-order-in-time variational formulation with exponential weights in time. We analyze the method, showing its stability without imposing any restrictions on the mesh size or time step, and proving quasi-optimal convergence for any choice of space-time tensor product discrete spaces that satisfies standard approximation assumptions. Numerical examples are provided to support the theoretical findings.