A new and improved analysis of the time domain boundary integral operators for acoustics

TL;DR

This paper introduces a novel, unified time-domain analysis of boundary integral operators for acoustics, simplifying proofs and extending applicability to various formulations and discretizations.

Contribution

It provides a new, general theorem for analyzing wave equation boundary operators using semi-group theory, improving upon previous results and simplifying the methodology.

Findings

Unified analysis framework for boundary integral operators

Improved regularity and well-posedness results

Applicability to multiple formulations and discretizations

Abstract

We present a novel analysis of the boundary integral operators associated to the wave equation. The analysis is done entirely in the time-domain by employing tools from abstract evolution equations in Hilbert spaces and semi-group theory. We prove a single general theorem from which well-posedness and regularity of the solutions for several boundary integral formulations can be deduced as particular cases. By careful choices of continuous and discrete spaces, we are able to provide a concise analysis for various direct and indirect formulations, both at the continuous level and for their Galerkin-in-space semi-discretizations. Some of the results here are improvements on previously known results, while other results are equivalent to those in the literature. The methodology presented here greatly simplifies the analysis of the operators of the Calderon projector for the wave equation and can be generalized for other relevant boundary integral equations.