Polynomial approximation of functions in Sobolev spaces

doi:10.1090/S0025-5718-1980-0559195-7

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Polynomial approximation of functions in Sobolev spaces

Journal Article•DOI•

Polynomial approximation of functions in Sobolev spaces

Todd F. Dupont¹, Ridgway Scott²•Institutions (2)

University of Chicago¹, University of Michigan²

01 Apr 1980-Mathematics of Computation (American Mathematical Society (AMS))-Vol. 34, Iss: 150, pp 441-463

TL;DR: In this paper, a polynomial plus a remainder is represented as a Taylor series and the remainder can be manipulated in many ways to give different types of bounds, including integer order and nonstandard Sobolev-like spaces.

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Abstract: Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.

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Book•

A Posteriori Error Estimation in Finite Element Analysis

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Mark Ainsworth¹, J. Tinsley Oden²•Institutions (2)

University of Leicester¹, University of Texas at Austin²

01 Jan 2000

TL;DR: In this paper, a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics is presented, focusing on methods for linear elliptic boundary value problems.

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Abstract: This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

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2,607 citations

Cites result from "Polynomial approximation of functio..."

...Such approximations have been considered by Clement [26] and Scott and Zhang [52]: THEOREM 1.1....
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...Dupont and Scott [60] obtained approximation results for piecewise polynomials based on regularized Taylor series expansions....
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Journal Article•DOI•

Finite element interpolation of nonsmooth functions satisfying boundary conditions

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L. Ridgway Scott, Shangyou Zhang

01 Apr 1990-Mathematics of Computation

TL;DR: In this article, a modified Lagrange type interpolation operator is proposed to approximate functions in Sobolev spaces by continuous piecewise polynomials, and the combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.

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Abstract: In this paper, we propose a modified Lagrange type interpolation operator to approximate functions in Sobolev spaces by continuous piecewise polynomials. In order to define interpolators for "rough" functions and to preserve piecewise polynomial boundary conditions, the approximated functions are averaged appropriately either on dor (d 1)-simplices to generate nodal values for the interpolation operator. This combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.

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1,648 citations

Journal Article•DOI•

Two families of mixed finite elements for second order elliptic problems

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Franco Brezzi¹, Jim Douglas², Luisa Donatella Marini•Institutions (2)

University of Pavia¹, University of Chicago²

01 Jun 1985-Numerische Mathematik

TL;DR: In this article, two families of mixed finite elements, one based on triangles and the other on rectangles, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces.

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Abstract: Two families of mixed finite elements, one based on triangles and the other on rectangles, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. Error estimates inL 2 (Ω) andH ?5 (Ω) are derived for these elements. A hybrid version of the mixed method is also considered, and some superconvergence phenomena are discussed.

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1,213 citations

Book Chapter•DOI•

Basic error estimates for elliptic problems

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Philippe G. Ciarlet¹•Institutions (1)

Pierre-and-Marie-Curie University¹

01 Jan 1991-Handbook of Numerical Analysis

1,097 citations

Journal Article•DOI•

Finite elements in computational electromagnetism

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Ralf Hiptmair¹•Institutions (1)

University of Tübingen¹

01 Jan 2002-Acta Numerica

TL;DR: In this paper, finite element Galerkin schemes for a number of linear model problems in electromagnetism were discussed, and the finite element schemes were introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms.

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Abstract: This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

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890 citations

Additional excerpts

...This permits us to continue the estimate by∥∥u−Π1pu∥∥L2(T ) ≤ Ch 1 2 T inf p∈(Pp( bT ))3 ‖û− p‖ Hs( bT ) ≤ Ch 1 2 T |û|Hs(bT ) ≤ Ch s T ‖u‖Hs(T ), where we used the Bramble–Hilbert lemma for fractional Sobolev norms on T̂ (Dupont and Scott 1980, Proposition 6.1) and Lemma 3.12....
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...…and find ‖u−Π1p|Tu‖ 2 L2(T ) ≤ C inf p∈(P0(bT ))3 ( ‖û− p‖2 Hs(bT ) + ‖curl û‖2 H s− 12 ( bT ) ) ≤ Cρ(Ωh)hT ( |û|2 Hs( bT ) + ‖curl û‖2 L2(bT ) + |curl û|2 H s− 12 ( bT ) ) , where a Bramble–Hilbert argument in fractional Sobolev spaces was invoked (Dupont and Scott 1980, Proposition 6.1)....
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References

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Book•

Singular Integrals and Differentiability Properties of Functions.

[...]

Elias M. Stein

01 Feb 1971

TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.

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Abstract: Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself. Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance. Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysis in 2005.

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9,595 citations

Journal Article•DOI•

Singular integrals and differentiability properties of functions

[...]

J. L. B. Cooper

01 Mar 1973-Bulletin of The London Mathematical Society

5,201 citations

Book•DOI•

Multiple integrals in the calculus of variations

[...]

Charles B. Morrey

01 Jan 2008

TL;DR: In this paper, a variational method in the theory of harmonic integrals has been proposed to solve the -Neumann problem on strongly pseudo-convex manifolds and parametric Integrals two-dimensional problems.

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Abstract: Semi-classical results.- The spaces Hmp and Hmp0.- Existence theorems.- Differentiability of weak solutions.- Regularity theorems for the solutions of general elliptic systems and boundary value problems.- A variational method in the theory of harmonic integrals.- The -Neumann problem on strongly pseudo-convex manifolds.- to parametric Integrals two dimensional problems.- The higher dimensional plateau problems.

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3,190 citations

"Polynomial approximation of functio..." refers background in this paper

...An inequality of the form (1.1) can be found in Morrey [16] (and implicitly in Sobolev [18]) for the case of P being all polynomials of degree at most r, A being all multi-indices of length r + 1, || • || being the norm on W™, and |-1 being the norm on L ....
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...The proofs of Bramble and Hubert used the results of Morrey and generalizations thereof....
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...1) can be found in Morrey [16] (and implicitly in Sobolev [18]) for the case of P being all polynomials of degree at most r, A being all multi-indices of length r + 1, || • || being the norm on WTM, and |-1 being the norm on L ....
[...]

Book•

Linear Partial Differential Operators

[...]

Lars Hörmander

01 Jan 1963

2,582 citations

"Polynomial approximation of functio..." refers background in this paper

...Let D(D) denote C (D) topologized with the usual inductive limit topology [13] ....
[...]

Book•

Lectures on Elliptic Boundary Value Problems

[...]

Shmuel Agmon¹•Institutions (1)

Hebrew University of Jerusalem¹

01 Jan 1965

TL;DR: In this article, the authors present an introduction to the theory of higher-order elliptic boundary value problems, and a detailed study of basic problems of the theory, such as the problem of existence and regularity of solutions of higher order elliptic edge value problems.

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Abstract: This book, which is a new edition of a book originally published in 1965, presents an introduction to the theory of higher-order elliptic boundary value problems. The book contains a detailed study of basic problems of the theory, such as the problem of existence and regularity of solutions of higher-order elliptic boundary value problems. It also contains a study of spectral properties of operators associated with elliptic boundary value problems. Weyl's law on the asymptotic distribution of eigenvalues is studied in great generality.

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1,568 citations

"Polynomial approximation of functio..." refers background in this paper

...Then SL v- <K^< c<"- «w- "■ * »)| |h(á) 4 ■ License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use APPROXIMATION OF FUNCTIONS IN SOBOLEV SPACES 453 Proof As is Agmon [1], it follows from Hubert's Nullstellensatz that there is an integer r such that for all |a| = r, k (5.1) Sa=?lRjXW,ß) for some polynomials RJ that are homogeneous of degree r - /.....
[...]
...Proof As is Agmon [1], it follows from Hubert's Nullstellensatz that there is an integer r such that for all |a| = r, k (5....
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