Error Analysis of Finite Element Methods With Triangles for Elliptic Boundary Value Problems

Barnhill, Robert E.; Whiteman, J. R.

doi:10.1016/b978-0-12-747250-8.50008-4

Cited by 17 publications

(5 citation statements)

References 11 publications

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“…One example of an interpolant for which the hypotheses of the Zero Kernel Theorem holds is given in the following theorem: Pro@ We show that P [xig(y)l is a polynomial in x of degree =i for the nontrivial case 0 ~ i < N, from which the conclusion follows. Now, generalizations of those given in 13arnhill and Whiteman [5]. Further details including calculations for quadratic interpolation on T can be found in Barnhill and Gregory [2].…”

Section: Error Bounds For Tensor Product Interpolationmentioning

confidence: 92%

Sard kernel theorems on triangular domains with application to finite element error bounds

Barnhill

Gregory

1975

Numer. Math.

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Summary. Error bounds for interpolation remainders on triangles are derived by means of extensions of the Sard Kernel Theorems. These bounds are applied to the Galerkin method for elliptic boundary value problems. Certain kernels are shown to be identically zero under hypotheses which are, for example, fulfilled by tensor product interpolants on rectangles. This removes certain restrictions on how the sides of the triangles and/or rectangles tend to zero. Explicit error bounds are computed for piecewise linear interpolation over a triangulation and applied to a model problem.

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Section: Error Bounds For Tensor Product Interpolationmentioning

confidence: 92%

Sard kernel theorems on triangular domains with application to finite element error bounds

Barnhill

Gregory

1975

Numer. Math.

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“…A need for introducing some singular elements as part of basis functions in certain finite element analysis arises out of the following considerations. The solution of certain problems, such as a field problem [1], exhibits highly singular behavior due to geometric features of the spatial domain. On the other hand, in other circumstances, the solution is overwhelmingly affected by the nature of loading and the problem of singularity can be ignored.…”

Section: Finite Element Methods With Singular Basis Functionsmentioning

confidence: 99%

A note on the finite element method with singular basis functions

Kaneko

Padilla

1999

Int. J. Numer. Meth. Engng.

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SUMMARYIn this note, we make a few comments concerning the paper of Hughes and Akin (Int. J. Numer. Meth. Engng., 15, 733-751 (1980)). Our primary goal is to demonstrate that the rate of convergence of numerical solutions of the ÿnite element method with singular basis functions depends upon the location of additional collocation points associated with the singular elements. Copyright ? 1999 John Wiley & Sons, Ltd.KEY WORDS: ÿnite element method with singular basis functions; the collocation method; singular interpolation elements FINITE ELEMENT METHOD WITH SINGULAR BASIS FUNCTIONSIn the paper [1], Hughes and Akin made an interesting observation concerning the ÿnite element analysis that incorporates singular element functions. A need for introducing some singular elements as part of basis functions in certain ÿnite element analysis arises out of the following considerations. The solution of certain problems, such as a ÿeld problem [2], exhibits highly singular behaviour due to geometric features of the spatial domain. On the other hand, in other circumstances, the solution is overwhelmingly a ected by the nature of loading and the problem of singularity can be ignored. To satisfy both situations just described, it is thought that an incorporation of singular elements that emulate the solution with the standard polynomial elements may perhaps be desirable. This is the point that was exploited by Hughes and Akin [1]. In order to make the computations of the ÿnite element method with singular elements more e cient, they consider the following algorithm for constructing interpolation functions. A construction of such algorithm was motivated by the idea that 'it is of practical interest to develop techniques for systematically deÿning shape functions for singularity modeling (and for developing special elements in general), which circumvent the interpolation problem ' [3; p:176]. The algorithm that they developed is as follows:

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“…En effet on peut calculer effectivement ~t partir des relations (2-10) des majorations de l'erreur d'interpolation d'Hermite dans la m6thode des ~l~ments finis droits. K = ~ et (~) 6tant d6finis comme dans la remarque 2-1, et les ~i~ v6rifiant (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) …”

Section: (~-X) -+(~-R)' L(u D)(x)unclassified

Estimation de l'erreur d'interpolation d'Hermite dans ? n

Gout

1977

Numer. Math.

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Error Analysis of Finite Element Methods With Triangles for Elliptic Boundary Value Problems

Cited by 17 publications

References 11 publications

Sard kernel theorems on triangular domains with application to finite element error bounds

Sard kernel theorems on triangular domains with application to finite element error bounds

A note on the finite element method with singular basis functions

Estimation de l'erreur d'interpolation d'Hermite dans ? n

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