A new approach to space-time boundary integral equations for the wave equation

TL;DR

This paper introduces a novel mathematical formulation for space-time boundary integral equations of the wave equation, enabling rigorous analysis and paving the way for improved numerical methods.

Contribution

The authors develop a new approach that proves continuity and inf-sup conditions for boundary integral operators in trace spaces, enhancing theoretical understanding and numerical analysis.

Findings

Operators are continuous and satisfy inf-sup conditions in trace spaces.

Framework supports sharper error estimates and adaptive methods.

Compatible with existing boundary element method implementations.

Abstract

We present a new approach for boundary integral equations for the wave equation with zero initial conditions. Unlike previous attempts, our mathematical formulation allows us to prove that the associated boundary integral operators are continuous and satisfy inf-sup conditions in trace spaces of the same regularity, which are closely related to standard energy spaces with the expected regularity in space and time. This feature is crucial from a numerical perspective, as it provides the foundations to derive sharper error estimates and paves the way to devise efficient adaptive space-time boundary element methods, which will be tackled in future work. On the other hand, the proposed approach is compatible with current time dependent boundary element method's implementations and we predict that it explains many of the behaviours observed in practice but that were not understood with the existing theory.