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A numerical method for a two-dimensional curl–curl and grad-div problem is studied in this paper. It is based on a discretization
using weakly continuous P
1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries. Optimal
convergence rates (up to an arbitrary positive ) in both the energy norm and the L
2 norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.
The work of the first author was supported in part by the National Science Foundation under Grant No. DMS-03-11790 and by
the Humboldt Foundation through her Humboldt Research Award. The work of the third author was supported in part by the National
Science Foundation under Grant No. DMS-06-52481. 相似文献
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In this paper we prove a family of inequalities for differential forms in Heisenberg groups H1 and H2, that are the natural counterpart of a class of div–curl inequalities in de Rham?s complex proved by Lanzani & Stein and Bourgain & Brezis. 相似文献
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M. Šilhavý 《Milan Journal of Mathematics》2008,76(1):275-306
The paper gives a decomposition of a general normal r-dimensional current [5] into the sum of three measures of which the first is an r-dimensional rectifiable measure, the second is the Cantor part of the current, and the third is Lebesgue absolutely continuous.
This is analogous to the well-known decomposition of the derivative of a function of bounded variation into the jump, Cantor,
and absolutely continuous parts; in fact the last is a special case of the result for (n–1)-dimensional normal currents. Further, Whitney’s cap product [15] is recast in the language of the approach to flat chains
by Federer [5] and a special case (viz., currents of dimension n – 1) is shown to be closely related to the measure-valued duality pairings between vector measures with curl a measure and
L∞ vectorfields with L∞ divergence as established by Anzellotti [2] and Kohn & Témam [6]. Finally, the cap product is shown to be jointly weak* continuous
in the two factors of the product in a way similar to the compensated compactness theory; in the cases of (n – 1)-dimensional objects this reduces to results closely related to the div–curl lemmas of the standard compensated compactness
theory.
Received: June 2007 相似文献
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In this note, we study a semilinear system involving the operator curl in an exterior domain in R3, which is the limiting form of the Ginzburg–Landau model for superconductors in three dimensions for a large value of the Ginzburg–Landau parameter. We prove that this problem has a smooth solution, and it decays exponentially at infinity. 相似文献
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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. 相似文献
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PARALLEL AUXILIARY SPACE AMG FOR H(curl) PROBLEMS 总被引:2,自引:0,他引:2
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. 相似文献
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A partially orthonormal basis is constructed with better conditioning properties for tetrahedral H(curl)-conforming Nédélec elements.The shape functions are cla... 相似文献
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In this article we investigate the analysis of a finite element method for solving H(curl; ??)-elliptic interface problems in general three-dimensional polyhedral domains with smooth interfaces. The continuous problems are discretized by means of the first family of lowest order Nédélec H(curl; ??)-conforming finite elements on a family of tetrahedral meshes which resolve the smooth interface in the sense of sufficient approximation in terms of a parameter ?? that quantifies the mismatch between the smooth interface and the triangulation. Optimal error estimates in the H(curl; ??)-norm are obtained for the first time. The analysis is based on a ??-strip argument, a new extension theorem for H 1(curl; ??)-functions across smooth interfaces, a novel non-standard interface-aware interpolation operator, and a perturbation argument for degrees of freedom for H(curl; ??)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution. 相似文献
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Inspection and maintenance of railway networks is a complex and expensive task. Special measurement vehicles are used to record the geometrical properties of railway lines within required time intervals. Due to the extent of measurement data the quality of railway track is evaluated considering only a few parameters. Although safety and comfort of wheel–rail–systems depend on the dynamical behavior, current inspection vehicles are not equipped to measure dynamic properties. In this paper, we will discuss a novel approach to evaluate the quality of railway tracks based on wheel–rail dynamics: Wheelset dynamics of subway trains are analyzed by Karhunen–Loève Transformation to extract the principal dynamics from the collected measurement data. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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Daniel J. Katz 《Designs, Codes and Cryptography》2012,65(3):291-324
We present a theorem that generalizes the result of Delsarte and McEliece on the p-divisibilities of weights in abelian codes. Our result generalizes the Delsarte–McEliece theorem in the same sense that the theorem of N. M. Katz generalizes the theorem of Ax on the p-divisibilities of cardinalities of affine algebraic sets over finite fields. As the Delsarte–McEliece theorem implies the theorem of Ax, so our generalization implies that of N. M. Katz. The generalized theorem gives the p-divisibility of the t-wise Hamming weights of t-tuples of codewords (c (1), . . . ,c (t)) as these words range over a product of abelian codes, where the t-wise Hamming weight is defined as the number of positions i in which the codewords do not simultaneously vanish, i.e., for which ${(c^{(1)}_i,\ldots,c^{(t)}_i)\not=(0,\ldots,0)}$ . We also present a version of the theorem that, for any list of t symbols s 1, . . . ,s t , gives p-adic estimates of the number of positions i such that ${(c^{(1)}_i,\ldots,c^{(t)}_i)=(s_1,\ldots,s_t)}$ as these words range over a product of abelian codes. 相似文献
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Song-lin Zhao 《Journal of Difference Equations and Applications》2013,19(8):1134-1148
We establish a discrete model for the potential Ablowitz–Kaup–Newell–Segur equation via a generalized Cauchy matrix approach. Soliton solutions and Jordan block solutions of this equation are presented by solving the determining equation set. By applying appropriate continuum limits, we obtain two semi-discrete potential Ablowitz–Kaup–Newell–Segur equations. The reductions to real and complex discrete and semi-discrete potential modified Korteweg-de Vries equations are also discussed. 相似文献
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In this article, we give a new proof of the Carey–Helton–Howe–Pincus trace formula using Kato's theory of “relatively-smooth” operators and Krein's trace formula. 相似文献
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Orlicz–Brunn–Minkowski type inequalities for Blaschke–Minkowski homomorphisms and their polars are established. 相似文献