An <i>hp</i>‐adaptive finite element method for electromagnetics. Part 3: A three‐dimensional infinite element for Maxwell's equations

Cecot, Witold; Rachowicz, Waldemar; Demkowicz, Leszek

doi:10.1002/nme.713

Cited by 35 publications

(13 citation statements)

References 30 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%

“…Note that (31) gives a variational form forê h which includes the approximation in (12). In the following we drop the subscript for the residual and the tangential jump, respectively, the localization will be shown when taking the norm (or some bilinear map) of these quantities.…”

Section: Implicit Error Estimate As a Lower Bound Of The Errormentioning

confidence: 99%

“…Examples of hp-adaptive techniques applied to the Maxwell equations can be found in e.g. [12,16,24,25,26,29]. The hp-adaptation technique is a promising approach to obtain efficient numerical algorithms to solve the Maxwell equations, but requires a reasonably accurate estimate of the local error in the numerical solution in order to control the adaptation process.…”

Section: Introductionmentioning

confidence: 99%

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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: Implicit Error Estimate As a Lower Bound Of The Errormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Implicit a posteriori error estimates for the Maxwell equations

2008

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“…When solving (possibly time-dependent) problems that may mix difficulties (corner singularities, boundary layer effects), hp adaptivity is required [14,28]. The elements can be subsected (isotropically and anisotropically) and their orders can be enriched which permits non-uniform distribution of element sizes h and orders p. Thus anisotropic polynomial degrees can be chosen in the computational domain, whereas h-refinement may lead to non-conforming meshes with hanging nodes that involve fewer degrees of freedom and thus gives smaller algebraic linear systems to solve than the ones we have obtained.…”

Section: Perspectivesmentioning

confidence: 99%

A numerical study on Neumann-Neumann methods forhpapproximations on geometrically refined boundary layer meshes II. Three-dimensional problems

Toselli¹,

Vasseur²

2006

ESAIM: M2AN

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Abstract.In this paper, we present extensive numerical tests showing the performance and robustness of a Balancing Neumann-Neumann method for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in three dimensions. The numerical results are in good agreement with the theoretical bound for the condition number of the preconditioned operator derived in [Toselli and Vasseur, IMA J. Numer. Anal. 24 (2004) 123-156]. They confirm that the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients. Good results are also obtained for certain singularly perturbed problems. The condition numbers only grow polylogarithmically with the polynomial degree, as in the case of p approximations on shape-regular meshes [Pavarino, RAIRO: Modél. Math. Anal. Numér. 31 (1997) 471-493]. This paper follows [Toselli and Vasseur, Comput. Methods Appl. Mech. Engrg. 192 (2003) IntroductionIn recent years hp finite element methods have gained an increasing popularity in both the applied mathematics community and some engineering application fields. The hp finite element method has been first introduced by Gui and Babuška [26] and since then some monographs [30,37,50,54] or part of textbooks ([38], Sect. 8.4), have proposed both theoretical and numerical insights into this topic. These methods are found to be particularly useful when high or extremely high accuracy is needed and when minimal dissipation and dispersion errors in the discrete system are required.Indeed the main reason for the interest in hp finite element methods is that they achieve exponential rates of convergence for both regular and singular solutions [37,50]. In presence of singularities or boundary layers, suitably graded meshes, geometrically refined towards corners, edges and/or faces have to be employed to achieve such an exponential rate of convergence [37,50]. Thus highly stretched meshes with huge aspect ratios are obtained in practice. Consequently, the condition number of the stiffness matrix severely deteriorates: [32] (see also the references therein). Unfortunately, up to now, no iterative substructuring method has been proven to be efficient when very thin elements and/or subdomains (involving meshes with high aspect ratio) or general non quasiuniform meshes are employed. Thus our goal is to fill this gap by proposing a domain decomposition preconditioner that is robust with the mesh aspect ratio and possible large jumps in the coefficients. Its theoretical derivation has been presented in [58]. In this paper the main emphasis is devoted to an extensive numerical study of its performances. These robustness aspects will be carefully summarized and analyzed hereafter for some three-dimensional elliptic model problems of diffusion or reaction-diffusion type. These model problems defined on simple geometries have been chosen here in order to be easily reproducible. Nevertheless some of them...

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“…There are several techniques to obtain explicit bounds for the unknown constant term (see e. g. [12]), but in most applications the estimates are somewhat pessimistic, hence the resulting estimators tend to be unrealistic and fail to detect the more subtle nuances of the specific problem. Several applications of adaptive methods with an explicit error estimation technique for the Maxwell equations can be found in [8,10,13,27,28].…”

Section: Introductionmentioning

confidence: 99%

Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates

Harutyunyan

Izsák

Vegt

et al. 2008

Computer Methods in Applied Mechanics and Engineering

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For the adaptive solution of the Maxwell equations on three-dimensional domains with Nédélec edge finite element methods, we consider an implicit a posteriori error estimation technique. On each element of the tessellation an equation for the error is formulated and solved with a properly chosen local finite element basis. We show that the discrete bilinear form of the local problems satisfies an inf-sup condition ensuring the well posedness of the error equations. An adaptive algorithm is developed based on the estimated error. We show that the method accurately predicts the regions in the domain with a larger error. The performance of the method is tested on various problems on non-convex domains with non-smooth boundaries. The numerical results show an accurate approximation of the true error. On the meshes generated adaptively with the help of the implicit a posteriori error estimation technique an error is obtained which is smaller than on globally refined meshes. Moreover, the convergence of the error on the locally adapted meshes is faster than that on the globally refined mesh.

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An hp‐adaptive finite element method for electromagnetics. Part 3: A three‐dimensional infinite element for Maxwell's equations

Abstract: SUMMARY

Cited by 35 publications

References 30 publications

Implicit a posteriori error estimates for the Maxwell equations

Implicit a posteriori error estimates for the Maxwell equations

A numerical study on Neumann-Neumann methods forhpapproximations on geometrically refined boundary layer meshes II. Three-dimensional problems

Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates

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