An Analysis of the Finite-Element Method

Strang, Gilbert; Fix, George J.; Griffin, D. S.

doi:10.1115/1.3423272

Cited by 539 publications

(719 citation statements)

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“…The interpolation-error (22) is thus seen to be of exactly the same order as the truncation error (14) [5], [19].…”

Section: From This Definition It Readily Follows Thatmentioning

confidence: 68%

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On a high order numerical method for functions with singularities

Eckhoff

1998

Math. Comp.

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Abstract. By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various types of singularities of importance in applications are considered. Relations between the discrete and the continuous Fourier series for the singular functions are established. Of particular interest are piecewise smooth functions, for which various important applications are indicated, and for which numerous numerical results are presented.

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“…The interpolation-error (22) is thus seen to be of exactly the same order as the truncation error (14) [5], [19].…”

Section: From This Definition It Readily Follows Thatmentioning

confidence: 68%

“…Finally, we would like to add that it is not difficult to find both one-and multi-dimensional problems which may require other types of special feature functions than the family U α described in this paper. The functions describing corner singularities [22] may here serve as illustrating examples.…”

Section: Discussionmentioning

confidence: 99%

On a high order numerical method for functions with singularities

Eckhoff

1998

Math. Comp.

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“…This observation was done by Strang and Fix [12] in the sense of a purely mathematical interpolation problem. What exactly means that the true field u and the interpolated field u h are such that in certain notable points ξ u (ξ ) = u h (ξ ).…”

Section: Lagrangian and Hermitian Finite Element Basismentioning

confidence: 99%

“…Historically, the higher order accuracy property was first noticed by Barlow [1] and its existence was firstly formulated by Strand and Fix [12]. This concept can be extended to similar properties but referring to O(h k+2 ) convergence and O(h k+3 ) convergence called simply ultraconvergence and hyperconvergence by using systematically the Taylor's Series Theorem [5,6].…”

Section: Definition Of Superconvergencementioning

confidence: 99%

Studies on Barlow points, Gauss points and Superconvergent points in 1D with Lagrangian and Hermitian finite element basis

Junior

Soares²

2008

Comput.appl.math.

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Abstract. In this work, we consider the superconvergence property of the finite element derivative for Lagrange's and Hermite's Family elements in the one dimensional interpolation problem. We also compare the Barlow points, Gauss points and Superconvergence points in the sense of Taylor's Series, confirming that they are not the same as believed before. We prove a not evident and new superconvergence property of Hermite's basis as well which shows that the centroid is not only a superconvergent for u h but an O(h 5 ) accuracy point.Mathematical subject classification: 74S05.

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“…Finite-element methods. Finite-element methods (FEMs) offer great flexibility in modeling problems with complex geometries and, as such, they have been widely used in science and engineering as the solvers of choice for structural, mechanical, heat transfer, and fluid dynamics problems [190,43,114,194,44,110,25].…”

Section: Methodsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

143

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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An Analysis of the Finite-Element Method

Cited by 539 publications

References 0 publications

On a high order numerical method for functions with singularities

On a high order numerical method for functions with singularities

Studies on Barlow points, Gauss points and Superconvergent points in 1D with Lagrangian and Hermitian finite element basis

A review of numerical methods for nonlinear partial differential equations

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