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NON SPURIOUS SPECTRAL-LIKE ELEMENT METHODS FOR MAXWELL'S EQUATIONS

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Journal of Computational Mathematics 2007年第3期25卷 282-304页

作者： Gary Cohen Marc Duruflé INRIA Domaine de VoluceauRocquencourt-BP 10578153 Le Chesnay CedexFrance

In this paper, we give the state of the art for the so called “mixed spectral elements” for Maxwell＇s equations. Several families of elements, such as edge elements and discon-tinuous Galerkin methods （DGM） are presented and discussed. In particular, we show the need of introducing some numerical dissipation terms to avoid spurious modes in these methods. Such terms are classical for DGM but their use for edge element methods is novel approach described in this paper. Finally, numerical experiments show the fast and low-cost character of these elements.

关键词： Maxwell's equations Discontinuous Galerkin methods edge elements Masslumping.

PARALLEL AUXILIARY SPACE AMG FOR H(curl) PROBLEMS

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Journal of Computational Mathematics 2009年第5期27卷 604-623页

作者： Tzanio V.Kolev Panayot S.Vassilevski Center for Applied Scientific Computing Lawrence Livermore National LaboratoryP.O.Box 808L-560LivermoreCA 94551USA

In this paper we review a number of auxiliary space based preconditioners for the second order definite and semi-definite Maxwell problems discretized with the lowest order Nedelec finite elements. We discuss the parallel implementation of the most promising of these methods, the ones derived from the recent Hiptmair-Xu （HX） auxiliary space decomposition [Hiptmair and Xu, SIAM J. Numer. Anal., 45 （2007）, pp. 2483-2509]. An extensive set of numerical experiments demonstrate the scalability of our implementation on large-scale H（curl） problems.

关键词： Parallel algebraic multigrid H(curl) problems edge elements Auxiliary space preconditioning.

ADAPTIVITY IN SPACE AND TIME FOR MAGNETOQUASISTATICS

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Journal of Computational Mathematics 2009年第5期27卷 642-656页

作者： Markus Clemens Jens Lang Delia Teleaga Georg Wimmer Helmut-Schmidt-Universitat Hamburg Holstenhofweg 8522043 HamburgGermany Technische Universitat Darmstadt Schlossgartenstraβe 764289 DarmstadtGermany

This paper addresses fully space-time adaptive magnetic field computations. We describe an adaptive Whitney finite element method for solving the magnetoquasistatic formulation of Maxwell＇s equations on unstructured 3D tetrahedral grids. Spatial mesh re- finement and coarsening are based on hierarchical error estimators especially designed for combining tetrahedral H（curl）-conforming edge elements in space with linearly implicit Rosenbrock methods in time. An embedding technique is applied to get efficiency in time through variable time steps. Finally, we present numerical results for the magnetic recording write head benchmark problem proposed by the Storage Research Consortium in Japan.

关键词： Magnetoquasistatics Space-time adaptivity edge elements Rosenbrock meth-ods Hierarchical error estimator SRC benchmark problem.

Local Multigrid in H(curl)

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Journal of Computational Mathematics 2009年第5期27卷 573-603页

作者： Ralf Hiptmair Weiying Zheng SAM ETH Zurich CH-8092 Ziirich Switzerland LSEC ICMSEC Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100190 China

We consider H（curl, Ω）-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H^1 （Ω）-context along with local discrete Helmholtz-type decompositions of the edge element space.

关键词： edge elements Local multigrid Stable multilevel splittings Subspace correc-tion theory Regular decompositions of H(curl ~) Helmholtz-type decompositions Local mesh refinement.

THREE-DIMENSIONAL NUMERICAL LOCALIZATION OF IMPERFECTIONS BASED ON A LIMIT MODEL IN ELECTRIC FIELD AND A LIMIT PERTURBATION MODEL

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Journal of Computational Mathematics 2009年第4期27卷 495-524页

作者： S.M.Mefire LAMFA CNRS UMR 6140Université de Picardie33 r.Saint-Leu80039 Amiens Cedex 1France

From a limit model in electric field obtained by letting the frequency vanish in the time-harmonic Maxwell equations, we consider a limit perturbation model in the tangential boundary trace of the curl of the electric field for localizing numerically certain small electromagnetic inhomogeneities, in a three-dimensional bounded domain. We introduce here two localization procedures resulting from the combination of this limit perturbation model with each of the following inversion processes： the Current Projection method and an Inverse Fourier method. Each localization procedure uses, as data, a finite number of boundary measurements, and is employed in the single inhomogeneity case; only the one based on an Inverse Fourier method is required in the multiple inhomogeneities case. Our localization approach is numerically suitable for the context of inhomogeneities that are not purely electric. We compare the numerical results obtained from the two localization procedures in the single inhomogeneity configuration, and describe, in various settings of multiple inhomogeneities, the results provided by the procedure based on an Inverse Fourier method.

关键词： Inverse problems Maxwell equations Electric fields Inhomogeneities Elec-trical Impedance Tomography Current Projection method FFT Numerical boundarymeasurements edge elements Least square systems Incomplete Modified Gram-Schmidt preconditioning.

STATIC REGIME IMAGING OF LOCATIONS OF CERTAIN 3D ELECTROMAGNETIC IMPERFECTIONS FROM A BOUNDARY PERTURBATION FORMULA

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Journal of Computational Mathematics 2014年第4期32卷 412-441页

作者： Seraphin M. Mefire IECL CNRS UMR 7502 Universite de Lorraine Faculte des Sciences et Technologies de Nancy 54506 Vandoeuvre-les-Nancy Cedex France

We are concerned, in a static regime, with a three-dimensional bounded domain of certain an imaging approach of the locations in electromagnetic imperfections. This approach is related to Electrical Impedance Tomography and makes use of a new perturbation formula in the electric fields. We present two localization procedures, from a Current Pro- jection method that deals with the single imperfection context and an inverse Fourier process that is devoted to multiple imperfections configurations. These procedures extend those that were described in our previous work, since operating for a broader class of settings. Namely, the localization is additionally performed for certain purely electric imperfections, as established from numerical simulations.

关键词： Inverse problems Maxwell equations Electric fields Inhomogeneities Elec-trical lmpedanee Tomography Current Projection method FFT Numerical boundarymeasurements Random noise edge elements Least square systems Incomplete ModifiedGram-Schmidt preconditioning.

Developing Finite Element Methods for Simulating Transformation Optics Devices with Metamaterials

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Communications in Computational Physics 2019年第1期25卷 135-154页

作者： Wei Yang Jichun Li Yungqing Huang Bin He Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105P.R.China Department of Mathematical Sciences University of Nevada Las VegasLas VegasNevada 89154-4020USA

In this paper,we first develop the mathematical modeling equations for wave propagation in several transformation optics devices,including electromagnetic concentrator,rotator and *** we propose the corresponding finite element time-domain methods for simulating wave propagation in these transformation optics *** implement the proposed algorithms and our numerical results demonstrate the effectiveness of our modeling *** our best knowledge,this is the first work on time-domain finite element simulation carried out for the electromagnetic concentrator,rotator and splitter.

关键词： Maxwell’s equations finite element time-domain methods edge elements transfor-mation optics

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell’s Equations

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Numerical Mathematics(Theory,Methods and Applications) 2012年第3期5卷 297-332页

作者： Ralf Hiptmair Haijun Wu Weiying Zheng SAM ETH ZürichCH-8092 ZürichSwizerland Department of Mathematics Nanjing UniversityNanjingJiangsu210093China LSEC Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing100190China

We consider the convergence theory of adaptive multigrid methods for secondorder elliptic problems and Maxwell’s *** multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their“immediate”*** the context of lowest order conforming finite element approximations,we present a unified proof for the convergence of adaptive multigrid V-cycle *** theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity *** convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of *** demonstrate our convergence theory by two numerical experiments.

关键词： MMaxwell’s equations Lagrangian finite elements edge elements adaptive multigrid method successive subspace correction