Some properties of layer potentials and boundary integral operators for the wave equation

Abstract

In this work we establish some new estimates for layer potentials of the acoustic wave equation in the time domain, and for their associated retarded integral operators. These estimates are proven using time-domain estimates based on theory of evolution equations and improve known estimates that use the Laplace transform.

The Costabel-Stephan system of boundary integral equations in the time domain

“…An exotic transmission problem. There is a form, based on [22,25,15], of writing the Galerkin semidiscretization (3.1)-(3.2) by focusing directly on (u h , v h ). Note that the potentials are defined on both sides of the interface Γ.…”

“…The analysis of the method follows recent techniques [26,15,5] entirely developed in the time domain, instead of the Laplace domain analysis that stems from the original work of Bamberger and Ha-Duong [2,3]. To the best of our knowledge, it is the first time that these techniques are used on a setting that does not involve a purely exterior boundary value problem.We note that part of the Laplace domain analysis for transmission problems using the Costabel-Stephan formulation was included in [22], and was extended recently to the case of electromagnetic waves in [10].…”

New mapping properties of the time domain electric field integral equation

“…Pure time domain analysis has been shown to outperform the double-back-through-Laplace-domain approach in several situations. Time domain analysis in this context originated in [23] as a tool to analyze long term stability of several boundary integral formulations in acoustics, and was developed in [12] to provide improved bounds for the retarded layer potentials and integral operators of transient acoustics. The same approach was further developed and refined for the direct integral equation formulation of transient scattering by a sound-soft obstacle [4], a boundary integral formulation for transmission problems in acoustics [20], and indirect formulations for Dirichlet and Neumann problems for acoustics [24].…”

“…What distinguishes this paper from the previous time-domain analysis is the transformation of the abstract second order differential equation associated to a dissipative operator to a system of first order equations (in time as well as in space). This simplifies the analysis with respect to [12,4,20] by avoiding the introduction of a cut-off boundary that was required to fit the problem in the correct functional framework. We note here that the techniques of [24] are not easy to adapt to the TDEFIE, due to non-minor complications in the associated Sobolev spaces.…”

Solving of initial-boundary value problems for the wave equation using retarded surface potential and Laguerre transform (in Ukrainian)

“…Вступ. Запiзнюючi потенцiали використовують для iнтегрального зображення кла-сичних ( [28]) i узагальнених ( [2,3,9]) розв'язкiв мiшаних задач для хвильового рiвняння в областях загального вигляду. Вони дають змогу замiнити мiшанi задачi для хвильово-го рiвняння еквiвалентними залежними вiд часової змiнної граничними iнтегральними рiвняннями (ЧГIР), у яких невiдомi величини -густини потенцiалiв -визначаються в кожен момент часу лише на межi областi ( [7,11,21,17,27]).…”