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Superconvergence of the gradient for the linear simplicial finite-elementmethod applied to elliptic equations is a well known featurein one, two, and three space dimensions. In this paper we showthat, in fact, there exists an elegant proof of this featureindependent of the space dimension. As a result, superconvergencefor dimensions four and up is proved simultaneously. The keyingredient will be that we embed the gradients of the continuouspiecewise linear functions into a larger space for which wedescribe an orthonormal basis having some useful symmetry properties.Since gradients and rotations of standard finite-element functionsare in fact the rotation-free and divergence-free elements ofRaviartThomas and Nédélec spaces in threedimensions, we expect our results to have applications alsoin those contexts. 相似文献
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Trivariate Cr macroelements defined in terms of polynomials of degree 8r + 1 on tetrahedra are analyzed. For r = 1,2, these spaces reduce
to well-known macroelement spaces used in data fitting and in the finite-element method.
We determine the dimension of these spaces, and describe stable local minimal determining sets and nodal minimal determining
sets. We also show that the spaces approximate smooth functions to optimal order. 相似文献
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We prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension, and design corresponding adaptive methods. The proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem, together with the Galerkin orthogonality of the finite-element method. The strong stability estimate uses the entropy condition for the system in an essential way. ©1995 John Wiley & Sons, Inc. 相似文献
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We consider a non-conforming domain decomposition techniquefor the discretization of the three-dimensional Stokes equationsby the mortar finite-element method. Relying on the velocitypressureformulation of the system, we perform the numerical analysisof residual error indicators for this problem and we prove thatthe error estimators provide upper and lower bounds for theenergy norm of the mortar finite-element solution. 相似文献
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The discrete Douglas problem: convergence results 总被引:1,自引:0,他引:1
We solve the problem of finding and justifying an optimal fullydiscrete finite-element procedure for approximating annulus-like,possibly unstable, minimal surfaces. In a previous paper weintroduced the general framework, obtained some preliminaryestimates, developed the ideas used for the algorithm, and gavenumerical results. In this paper we prove convergence estimates. 相似文献
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Dimitrios Ballas 《代数通讯》2017,45(2):481-492
In this paper, we show that the injective dimension of all projective modules over a countable ring is bounded by the self-injective dimension of the ring. We also examine the extent to which the flat length of all injective modules is bounded by the flat length of an injective cogenerator. To that end, we study the relation between these finiteness conditions on the ring and certain properties of the (strict) Mittag–Le?er modules. We also examine the relation between the self-injective dimension of the integral group ring of a group and Ikenaga’s generalized (co-)homological dimension. 相似文献
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在这篇文章里,我们证明了,当环S是R的excellent扩张,M是S-模时,M做为S-模的弱维数与M做为R-模的弱维数相等。 相似文献
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We consider a problem of elliptic optimal design. The control is the shape of the domain on which the Dirichlet problem for the Laplace equation is posed. In dimension n=2, S?veràk proved that there exists an optimal domain in the class of all open subsets of a given bounded open set, whose complements have a uniformly bounded number of connected components. The proof (J. Math. Pures Appl. 72 (1993) 537–551) is based on the compactness of this class of domains with respect to the complementary-Hausdorff topology and the continuous dependence of the solutions of the Dirichlet Laplacian in H1 with respect to it. In this Note we consider a finite-element discrete version of this problem and prove that the discrete optimal domains converge in that topology towards the continuous one as the mesh-size tends to zero. The key point of the proof is that finite-element approximations of the solution of the Dirichlet Laplacian converge in H1 whenever the polygonal domains converge in the sense of that topology. To cite this article: D. Chenais, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004). 相似文献
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Josef Dalík 《Applications of Mathematics》2008,53(6):547-560
We study the problem of Lagrange interpolation of functions of two variables by quadratic polynomials under the condition
that nodes of interpolation are vertices of a triangulation. For an extensive class of triangulations we prove that every
inner vertex belongs to a local six-tuple of vertices which, used as nodes of interpolation, have the following property:
For every smooth function there exists a unique quadratic Lagrange interpolation polynomial and the related local interpolation
error is of optimal order. The existence of such six-tuples of vertices is a precondition for a successful application of
certain post-processing procedures to the finite-element approximations of the solutions of differential problems.
This work was supported by the grant GA ČR 103/05/0292. 相似文献