A Time-Domain Finite Element Method for Helmholtz Equations

Van, Tri; Wood, Aihua W.

doi:10.1006/jcph.2002.7204

Cited by 29 publications

(17 citation statements)

References 7 publications

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“…Convergence properties of the finite element scheme for the fully discrete problem are also discussed. Numerical experiments are performed for two-dimensional cavity problems that demonstrate the accuracy and stability of the method and are reported in [24].…”

Section: Discussionmentioning

confidence: 45%

A Time-Domain Finite Element Method for Maxwell's Equations

Van¹,

Wood²

2004

SIAM J. Numer. Anal.

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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

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Section: Discussionmentioning

confidence: 45%

A Time-Domain Finite Element Method for Maxwell's Equations

Van¹,

Wood²

2004

SIAM J. Numer. Anal.

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“…In the context of design, an optimization procedure was used to enhance or reduce the RCS in [7,8]. A great deal of mathematical and computational methods have been studied for the scattering by cavities, including variational and finite element methods ( [2,4]), integral equation methods ( [1,3,5,25]), time-domain methods ( [21][22][23][24]), and hybrid methods ( [6,10,15,16,19,20]). The goal of this work is to establish the stability estimates of cavity problems with explicit dependence on the high wavenumber.…”

Section: Introductionmentioning

confidence: 37%

“…Analogous to (22), the estimate containing the auxiliary term resulting from the assumption of this lemma takes the following form:…”

Section: Proofmentioning

confidence: 48%

Stability for the Electromagnetic Scattering from Large Cavities

Bao

Yun

2015

Arch Rational Mech Anal

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Consider the scattering of electromagnetic waves from a large rectangular cavity embedded in the infinite ground plane. There are two fundamental polarizations for the scattering problem in two dimensions: TM (transverse magnetic) and TE (transverse electric). In this paper, new stability results for the cavity problems are established for large rectangular shape cavities in both polarizations. For the TM cavity problem, an asymptotic property of the solution and a stability estimate with an improved dependence on the high wavenumber are derived. In the TE case, the first stability result is established with an explicit dependence on the wave number.

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“…We overcome this difficulty by the use of collectively compact operators. Numerical results based on the method developed here is reported in [24].…”

Section: Introductionmentioning

confidence: 42%

Finite element analysis of transient electromagnetic scattering problems

Van¹,

Wood

2005

Adv Comput Math

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In this paper, Newmark time-stepping scheme and edge elements are used to numerically solve the time-dependent scattering problem in a three-dimensional cavity. Finite element methods based on the variational formulation derived in [23] are considered. Due to the lack of regularity of ε r , the existence and uniqueness of the discrete solutions and their convergence are proved by using the concept of collectively compact operators. An optimal convergence rate in the energy norm is also established.

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A Time-Domain Finite Element Method for Helmholtz Equations

Cited by 29 publications

References 7 publications

A Time-Domain Finite Element Method for Maxwell's Equations

A Time-Domain Finite Element Method for Maxwell's Equations

Stability for the Electromagnetic Scattering from Large Cavities

Finite element analysis of transient electromagnetic scattering problems

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