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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 $[ - 1,1]$ with elements from the null-space of ${P^N}$, $N \geqslant 1$, 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.
- Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI https://doi.org/10.1007/BF02165003
- I. Babuška, B. A. Szabo, and I. N. Katz, The $p$-version of the finite element method, SIAM J. Numer. Anal. 18 (1981), no. 3, 515–545. MR 615529, DOI https://doi.org/10.1137/0718033
- M. S. Baouendi and C. Goulaouic, Régularité et théorie spectrale pour une classe d’opérateurs elliptiques dégénérés, Arch. Rational Mech. Anal. 34 (1969), 361–379 (French). MR 249844, DOI https://doi.org/10.1007/BF00281438 J. Bergh & J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin and New York, 1976. P. Destuynder, Sur une Justification des Modeles de Plaques et de Coques par les Méthodes Asymptotiques, Doctoral Thesis, Paris VI, January 1980.
- V. K. Dzyadyk, Vvedenie v teoriyu ravnomernogo priblizheniya funktsiĭ polinomami, Izdat. “Nauka”, Moscow, 1977 (Russian). MR 0612836 D. G. Gordeziani, "Accuracy of a variant in the theory of thin shells," Soviet Phys. Dokl., v. 19, 1974, pp. 385-386.
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177
- G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785 S. M. Nikol’skiĭ, Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, Berlin and New York, 1975. F. G. Tricomi, Differential Equations, Blackie & Son, London, 1961. M. Vogelius, A Dimensional Reduction Approach to the Solution of Partial Differential Equations, Ph. D. Thesis, University of Maryland, December 1979.
- M. Vogelius and I. Babuška, On a dimensional reduction method. I. The optimal selection of basis functions, Math. Comp. 37 (1981), no. 155, 31–46. MR 616358, DOI https://doi.org/10.1090/S0025-5718-1981-0616358-0
Retrieve articles in Mathematics of Computation with MSC: 65N99, 65J10
Retrieve articles in all journals with MSC: 65N99, 65J10
Additional Information
Article copyright:
© Copyright 1981
American Mathematical Society