Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
MathSciNet review: 616359
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • [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)

Similar Articles

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

American Mathematical Society