On a dimensional reduction method. II. Some approximation-theoretic results

Authors:
M. Vogelius and I. Babuška

Journal:
Math. Comp. **37** (1981), 47-68

MSC:
Primary 65N99; Secondary 65J10

DOI:
https://doi.org/10.1090/S0025-5718-1981-0616359-2

MathSciNet review:
616359

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Abstract: This paper is the second in a series of three that analyze a method of dimensional reduction. It contains some results for approximation of functions on the interval with elements from the null-space of , , where *P* is a second-order ordinary differential operator. A special case of this is approximation by polynomials.

The one-dimensional results are used as a tool to prove similar versions in several dimensions. These multi-dimensional results are directly related to the approximate method of dimensional reduction that was introduced in [13], and they lead to statements about the convergence properties of this approach.

The third paper, which analyzes the adaptive aspects of the method, is forthcoming.

**[1]**I. Babušska, "Error-bounds for finite element method,"*Numer. Math.*, v. 16, 1970/71, pp. 322-333. MR**0288971 (44:6166)****[2]**I. Babušska, B. A. Szabo & I. N. Katz,*The p-Version of the Finite Element Method*, Report WU/CCM-79/1, Washington University in St. Louis, May 1979;*SIAM J. Numer. Anal.*(To appear.) MR**615529 (82j:65081)****[3]**M. S. Baouendi & C. Goulaouic, "Régularité et théorie spectrale pour une classe d'opérateurs elliptiques dégénères,*Arch. Rational Mech. Anal.*, v. 34, 1969, pp. 361-379. MR**0249844 (40:3085)****[4]**J. Bergh & J. Löfström,*Interpolation Spaces*, Springer-Verlag, Berlin and New York, 1976.**[5]**P. Destuynder,*Sur une Justification des Modeles de Plaques et de Coques par les Méthodes Asymptotiques*, Doctoral Thesis, Paris VI, January 1980.**[6]**V. K. Dzjadyk,*An Introduction to the Theory of Uniform Approximation of Functions by Polynomials*, "Nauka", Moscow, 1977. (Russian) MR**0612836 (58:29579)****[7]**D. G. Gordeziani, "Accuracy of a variant in the theory of thin shells,"*Soviet Phys. Dokl.*, v. 19, 1974, pp. 385-386.**[8]**J. L. Lions & E. Magenes,*Non-Homogeneous Boundary Value Problems and Applications*, Vol. I, Springer-Verlag, Berlin and New York, 1972. MR**0350177 (50:2670)****[9]**G. G. Lorentz,*Approximation of Functions*, Holt, Rinehart & Winston, New York, 1966. MR**0213785 (35:4642)****[10]**S. M. Nikol'skiĭ,*Approximation of Functions of Several Variables and Imbedding Theorems*, Springer-Verlag, Berlin and New York, 1975.**[11]**F. G. Tricomi,*Differential Equations*, Blackie & Son, London, 1961.**[12]**M. Vogelius,*A Dimensional Reduction Approach to the Solution of Partial Differential Equations*, Ph. D. Thesis, University of Maryland, December 1979.**[13]**M. Vogelius & I. Babuška. "On a dimensional reduction method. I. The optimal selection of basis functions,"*Math Comp.*, v. 37, 1981, pp. 31-46. MR**616358 (83c:65259a)**

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DOI:
https://doi.org/10.1090/S0025-5718-1981-0616359-2

Article copyright:
© Copyright 1981
American Mathematical Society