New mapping properties of the time domain electric field integral equation

Abstract

We show some improved mapping properties of the Time Domain Electric Field Integral Equation and of its Galerkin semidiscretization in space. We relate the weak distributional framework with a stronger class of solutions using a group of strongly continuous operators. The stability and error estimates we derive are sharper than those in the literature.Proof. This is a straightforward consequence of the definition of G h and of the characterization of the Maxwell single layer potential by the transmission problem (3.2). Note that the second equation in (3.9) can be equivalently written as a Galerkin semidiscreteProposition 3.4. Let J ∈ TD(H −1/2 (div Γ , Γ)) and define

A new and improved analysis of the time domain boundary integral operators for the acoustic wave equation

“…The direct-in-time study of the acoustic Calderón projector began in [17] and was detailed in the second part of [18], employing a second order (in time and in space) equation approach, namely, the problems were rewritten as a second-order-in-time differential equation associated to an unbounded (second order differential) operator in the space variables. This approach later proved to be inflexible for the treatment of Maxwell equations, this lead to the use of semigroup theory in [16], greatly simplifying the analysis and sidestepped the cut-off process and reconciliation step described in [18,4,15]. Moreover, the estimates obtained with the direct-in-time analysis are sharper than those obtained through Laplace domain analysis.…”

“…• The application of these ideas to Maxwell equations actually precedes this paper [16], given the fact that the second order equation ideas [17,18] seem not to apply to the functional space setting of the layer potentials for electromagnetism.…”

Time-domain boundary integral equation modeling of heat transmission problems

“…What we do in this paper is related to the purely time-domain analysis of TDBIE initiated in [17] for wave propagation problems. This theory has seen different extensions and refinements: for instance, [37] extends the results to the Maxwell equations and [22] is the realization that a first order in time formulation makes the analysis much simpler. The goal of exploring purely time-domain techniques is multiple.…”