Finite Element Methods for
            <scp>M</scp>
            axwell Equations

Demkowicz, Leszek

doi:10.1002/0470091355.ecm023

Cited by 11 publications

(10 citation statements)

References 48 publications

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“…More specifically, recall that in Section 2, the weak form (9) for the local error equation equipped with the natural boundary condition (12) results in the variational problem (15) which is well posed by Lemma 2 if and only if k is not an eigenvalue of the appropriate boundary value problem (18). In this section, we determine the eigenvalues belonging to cubic subdomains.…”

Section: Lemmamentioning

confidence: 98%

“…The quantity γ τ curlE will be called the natural boundary data from now on. In the literature, the homogeneous natural boundary condition is called the magnetic symmetry wall condition [15]. Introducing this approximation into (9) we arrive at the variational problem for the implicit error estimate: Find…”

Section: Formulation Of the Local Problemmentioning

confidence: 99%

“…For a short, self-contained introduction to finite element methods for the Maxwell equations we refer to [15].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Implicit a posteriori error estimates for the Maxwell equations

2008

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Abstract. An implicit a posteriori error estimation technique is presented and analyzed for the numerical solution of the time-harmonic Maxwell equations using Nédélec edge elements. For this purpose we define a weak formulation for the error on each element and provide an efficient and accurate numerical solution technique to solve the error equations locally. We investigate the well-posedness of the error equations and also consider the related eigenvalue problem for cubic elements. Numerical results for both smooth and non-smooth problems, including a problem with reentrant corners, show that an accurate prediction is obtained for the local error, and in particular the error distribution, which provides essential information to control an adaptation process. The error estimation technique is also compared with existing methods and provides significantly sharper estimates for a number of reported test cases.

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Section: Lemmamentioning

confidence: 98%

Section: Formulation Of the Local Problemmentioning

confidence: 99%

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Implicit a posteriori error estimates for the Maxwell equations

2008

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“…It supports geometrical refinements through the use of its own geometrical package to accurately model the problem geometry. Hexahedral H(curl) conforming finite elements satisfying the commutativity of the so-called de Rham diagram and hence, assuring convergence and stability of the numerical solution, are used, [8,7].…”

Section: Hp-fem and Automatic Hp-adaptivity In 3dmentioning

confidence: 99%

High-accuracy adaptive modeling of the energy distribution of a meniscus-shaped cell culture in a Petri dish

Gomez-Revuelto

Garcı́a-Castillo

Pardo

2015

Journal of Computational Science

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Cylindrical Petri dishes embedded in a rectangular waveguide and exposed to a polarized electromagnetic wave are often used to grow cell cultures. To guarantee the success of these cultures, it is necessary to enforce that the specific absorption rate distribution is sufficiently high and uniform over the Petri dish. Accurate numerical simulations are needed to design such systems. These simulations constitute a challenge due to the strong discontinuity of electromagnetic material properties involved, the relative low field value within the dish cultures compared with the rest of the domain, and the presence of the meniscus shape developed at the liquid boundary. The latter greatly increases the level of complexity of the model in terms of geometry and intensity of the gradients/singularities of the field solution. In here, we employ a three-dimensional (3D) hp-adaptive finite element method using isoparametric elements to obtain highly accurate simulations. We analyze the impact of the geometrical modeling of the meniscus shape cell culture in the hp-adaptivity. Numerical results showing the error convergence history indicate the numerical difficulties arisen due to the presence of a meniscus-shaped object. At the same time, the resulting energy distribution shows that to consider such meniscus shape is essential to guarantee the success of the cell culture from the biological point of view.

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“…With the electric field vanishing in a perfect conductor, and the tangential component of E being continuous across material interfaces, the tangential component of E on a boundary adjacent to a perfect conductor must vanish [8] n ×Ê φ = 0. In our case this simply translates into the condition,…”

Section: Statement Of the Problemmentioning

confidence: 99%

Verification of Goal-Oriented HP-Adaptivity

Paszynski¹,

Demkowicz²,

Pardo³

2005

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The paper presents calculations for the radiation of a loop antenna wrapped around a metallic cylinder into a conductive medium. The problem has been solved analytically by using Fourier transform and Bessel functions. The evaluation of the analytical solution, involves the use of exponentially scaled Bessel functions, and FFT to evaluate inverse Fourier transform, with all computations done in MATLAB. The analytical solution has been used to validate a goaloriented hp adaptive strategy based on a simultaneous solution of a dual problem, and an hp strategy based on minimizing the projection-based interpolation error of a reference solution.

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Finite Element Methods for M axwell Equations

Cited by 11 publications

References 48 publications

Implicit a posteriori error estimates for the Maxwell equations

Implicit a posteriori error estimates for the Maxwell equations

High-accuracy adaptive modeling of the energy distribution of a meniscus-shaped cell culture in a Petri dish

Verification of Goal-Oriented HP-Adaptivity

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