Very Weak Space-Time Variational Formulation for the Wave Equation: Analysis and Efficient Numerical Solution

TL;DR

This paper develops a novel very weak space-time variational formulation for the wave equation, providing a stable and efficient numerical method with strong error control and practical solvers, especially effective in low regularity scenarios.

Contribution

It introduces a new weak variational formulation and a tensor product Petrov-Galerkin discretization with optimal stability, enabling efficient and accurate solutions for wave equations.

Findings

The formulation is well-posed even with minimal regularity.

The discretization achieves optimal inf-sup stability.

Numerical experiments demonstrate competitive performance in low regularity cases.

Abstract

We introduce a very weak space-time variational formulation for the wave equation, prove its well-posedness (even in the case of minimal regularity) and optimal inf-sup stability. Then, we introduce a tensor product-style space-time Petrov-Galerkin discretization with optimal discrete inf-sup stability, obtained by a non-standard definition of the trial space. As a consequence, the numerical approximation error is equal to the residual, which is particularly useful for a posteriori error estimation. For the arising {discrete linear systems} in space and time, we introduce efficient numerical solvers that appropriately exploit the equation structure, either at the preconditioning level or in the approximation phase by using a tailored Galerkin projection. This Galerkin method shows competitive behavior concerning {wall-clock} time, accuracy and memory as compared with a standard time-stepping method in particular in low regularity cases. Numerical experiments with a 3D (in space) wave equation illustrate our findings.