Uniform error estimates for certain narrow Lagrange finite elements

Shenk, Norman

doi:10.1090/s0025-5718-1994-1226816-5

Cited by 22 publications

(3 citation statements)

References 5 publications

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“…Results for the standard finite element approximations in three dimensions were obtained in several papers (see, for example, [2,6,8,10,13]). In [10] Krízek proved optimal order error estimates for the Lagrange interpolation on a tetrahedron K for smooth functions, namely u ∈ W 2,∞ , provided the angles between faces and the angles in the faces are bounded away from π.…”

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Error Estimates for the Raviart–Thomas Interpolation Under the Maximum Angle Condition

Durán¹,

Lombardi²

2008

SIAM J. Numer. Anal.

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The classical error analysis for the Raviart-Thomas interpolation on triangular elements requires the so-called regularity of the elements, or equivalently, the minimum angle condition. However, in the lowest order case, optimal order error estimates have been obtained in [G. Acosta and R. G. Durán, SIAM J. Numer. Anal., 37 (2000), pp. 18-36] replacing the regularity hypothesis by the maximum angle condition, which was known to be sufficient to prove estimates for the standard Lagrange interpolation. In this paper we prove error estimates on triangular elements for the Raviart-Thomas interpolation of any order under the maximum angle condition. Also, we show how our arguments can be extended to the three-dimensional case to obtain error estimates for tetrahedral elements under the regular vertex property introduced in 1. Introduction. The classical error analysis for finite element approximations is based on the so-called regularity assumption on the elements. In other words, the constants in the error estimates obtained depend on the ratio between outer and inner diameter of the elements and blow up when this ratio goes to infinity (see, for example, [4,5]).However, it is well known that the regularity assumption can be relaxed for standard finite element approximations. For example, in the two-dimensional (2d) case, optimal order error estimates have been proved for triangular elements under the weaker maximum angle condition (i.e., angles bounded away from π). This condition allows the use of the so-called anisotropic elements which is of interest in several applications, for example in problems with boundary or interior layers.Error estimates under the maximum angle condition were first obtained in [3,9]. After these pioneering works many papers have considered different generalizations of their results (see, for example, [2] and the references therein).The usual error analysis for mixed finite element methods also makes use of the regularity assumption [12,14]. In view of the results for the standard method mentioned above, it is a natural question whether the regularity hypothesis on the elements can be relaxed in this case also. A positive answer for the Raviart-Thomas space of lowest order RT 0 is given in [1], where an optimal error estimate is proved under the maximum angle condition. However, it is not straightforward to extend the arguments given in that paper to higher order approximations. In [7] it is proved that the maximum angle condition is also sufficient to obtain optimal error estimates for the Raviart-Thomas space RT 1 .

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Error Estimates for the Raviart–Thomas Interpolation Under the Maximum Angle Condition

Durán¹,

Lombardi²

2008

SIAM J. Numer. Anal.

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“…The literature on anisotropic finite elements is nowadays rich and split basically into two categories: (i) the analysis of approximation properties of such elements for "regular" solutions, but under minimal requirements on the mesh (see [36], [2], [29], [1]); (ii) the analysis of approximation properties of "singular" solutions on suitably designed meshes, and with the aim of recovering algebraic convergence for finite elements of any order. This approach goes back to [8] and was developed in dimension three for low order Lagrangian finite elements in [4] (see also [7]), and partially extended to low order finite element approximation for vector problems in [30], [40] under restrictive assumptions on the geometry of the polyhedron .…”

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Algebraic convergence for anisotropic edge elements in polyhedral domains

2005

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We study approximation errors for the h-version of Nédélec edge elements on anisotropically refined meshes in polyhedra. Both tetrahedral and hexahedral elements are considered, and the emphasis is on obtaining optimal convergence rates in the H(curl) norm for higher order elements. Two types of estimates are presented: First, interpolation error estimates for functions in anisotropic weighted Sobolev spaces. Here we consider not only the H(curl)-conforming Nédélec elements, but also the H(div)-conforming Raviart-Thomas elements which appear naturally in the discrete version of the de Rham complex. Our technique is to transport error estimates from the reference element to the physical element via highly anisotropic coordinate transformations. Second, Galerkin error estimates for the standard H(curl) approximation of time harmonic Maxwell equations. Here we use the anisotropic weighted Sobolev regularity of the solution on domains with three-dimensional edges and corners. We also prove the discrete compactness property needed for the convergence of the Maxwell eigenvalue problem. Our results generalize those of [40] to the case of polyhedral corners and higher order elements.

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“…Since the end of eighties of the last century, the anisotropic elements have been extensively studied. In particular, for the interpolation error estimate we refer to Acosta [5], Apel [6,7], Chen [8,9], D'Azevedo [10], Duran [11,12], Kȓízek [13,14], Rippa [15], Shenk [16], Zenisek [17,18] and references therein. The studies on the anisotropic nonconforming elements are quite new, there are only a few papers related to this topic (cf.…”

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Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

Mao

Shi

2006

SCI CHINA SER A

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In this paper, we consider the nonconforming rotated Q 1 element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.

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Uniform error estimates for certain narrow Lagrange finite elements

Cited by 22 publications

References 5 publications

Error Estimates for the Raviart–Thomas Interpolation Under the Maximum Angle Condition

Error Estimates for the Raviart–Thomas Interpolation Under the Maximum Angle Condition

Algebraic convergence for anisotropic edge elements in polyhedral domains

Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

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