In this paper, we propose a novel scheme to accelerate integral equation solvers when applied to multiscale problems. These class of problems exhibit multiple length/frequency scales and arise when analyzing scattering/radiation from realistic structures where dense discretization is necessary to accurately capture geometric features. Solutions to the discretized integral equations due to these structures is challenging, due to their high computational cost and ill-conditioning of the resulting matrix system. The focus of this paper is on ameliorating the computational cost. Our approach will rely on exploiting the recently developed accelerated Cartesian expansion (ACE) algorithm to arrive at a method that is stable and efficient at low frequencies. These will then be integrated with the well known fast multipole method, thus forming a scheme that is wideband. Rigorous convergence estimates of this method are derived, and convergence and efficiency of the overall fast method is demonstrated. These are then integrated into an existing integral equation solver, whose efficiency is demonstrated for some practical problems. Index Terms-Accelerated Cartesian expansion (ACE), Cartesian expansions, fast multipole method (FMM), fast solvers, integral equation (IE), multipole methods, multiscale, scattering, wideband.

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SUMMARYIn this paper, Newmark time-stepping scheme and edge elements are used to numerically solve the time-dependent scattering problem in a three-dimensional polyhedral cavity. Finite element methods based on the variational formulation derived in Van and Wood (Adv. Comput. Math., to appear) are considered. Existence and uniqueness of the discrete problem is proved by using Babu ska-Brezzi theory. Finite element error estimate and stability of the Newmark scheme are also established.

Presented here is a time-domain finite element method for approximating Maxwell's equations. The problem is to approximate the electromagnetic fields scattered by a bounded, inhomogeneous cavity embedded in an infinite ground plane. The time-dependent scattering problem is first discretized in time by Newmark's time-stepping scheme. The resulting semidiscrete problem is proved to be well posed. A nonlocal boundary condition on the cavity aperture is constructed to reduce the computational domain to the cavity itself. Stability analysis and error estimates of the fully discrete problem are provided. Introduction.Time-harmonic (frequency-domain) Maxwell's equations for scattering problems are well studied and documented ([1, 19, 20, 21], to name a few) compared to their time-domain counterparts. This is due, to a large extent, to their obvious advantage of the absence of the time dependence and the limitations of computer power. The recent rapid advances in computing technology have prompted a growing popularity of numerical schemes for simulating electromagnetic transients (time-domain) for their potential to generate wide-band data and model nonlinear materials. Reports of new and faster numerical techniques for electromagnetic analysis have flourished in the engineering literature (see, for example, [18,23,12,25]). However, very little analysis is known in the open literature. To the authors' knowledge, the first mathematical study of time-domain Maxwell's equations for scattering by a bounded perfectly electric conducting (PEC) body was reported in [22], in which a spatially discretized problem is analyzed. More recently, in [4,5], fully discrete finite element methods for solving Maxwell's equations of bounded PEC scattering bodies are considered. In both cases, the problem is defined in a bounded domain. This paper serves as our first attempt to understand the various stability and convergence issues associated with the finite element method for modelling transient electromagnetic scattering from non-PEC bodies. The problem is defined in an infinite space.As observed in [7], most numerical schemes for scattering problems are faced with the problem of truncating the infinite domain to a bounded computational domain