Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations


Author: Stephen Hilbert
Journal: Math. Comp. 27 (1973), 81-89
MSC: Primary 65D10; Secondary 65N99
DOI: https://doi.org/10.1090/S0025-5718-1973-0331715-3
MathSciNet review: 0331715
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a given uniform grid of $ {E^N}$ (N-dimensional Euclidean space) with mesh h, a class of smoothing functions (mollifiers) is constructed. If a function is an element of the Sobolev space $ H_2^m$, then the error made by replacing the given function by a smoother $ ({C^\infty })$ function (which is the given function convolved with one of the mollifiers) is bounded by a constant times $ {h^m}$.

This result is used to construct approximations for functions using Hermite or spline interpolation, even though the function to be approximated need not satisfy the continuity conditions necessary for the existence of a Hermite or spline interpolate. These techniques are used to find approximations to the generalized solution of a second order elliptic Neumann problem.


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

  • [1] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Math. Studies, no. 2, Van Nostrand, Princeton, N. J., 1965. MR 31, #2504. MR 0178246 (31:2504)
  • [2] J. Bramble & S. Hilbert, "Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation," SIAM J. Numer. Anal., v. 7, 1970, pp. 112-124. MR 41 #7819. MR 0263214 (41:7819)
  • [3] J. Bramble & S. Hilbert, "Bounds for a class of linear functionals with applications to Hermite interpolation," Numer. Math., v. 16, 1971, pp. 362-369. MR 0290524 (44:7704)
  • [4] J. Bramble & A. Schatz, "Least squares methods for 2mth order eliptic boundary-value problems," Math. Comp., v. 25, 1971, pp. 1-32. MR 0295591 (45:4657)
  • [5] J. L. Lions, Problèmes aux limites dans les équations aux dérivées partielles, 2ième éd., Séminaire de Mathématiques Supérieures, no. 1 (Été, 1962), Les Presses de l'Université de Montréal, Montreal, Que., 1965. MR 40 #4602. MR 0251372 (40:4602)
  • [6] J. Nitsche, "Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens," Numer. Math., v. 11, 1968, pp. 346-348. MR 38 #1823. MR 0233502 (38:1823)
  • [7] R. Varga, "Functional analysis and approximation theory in numerical analysis," Proceedings of the regional N.S.F. Conference at Boston Univ., July 20-24, 1970. MR 0310504 (46:9602)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D10, 65N99

Retrieve articles in all journals with MSC: 65D10, 65N99


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1973-0331715-3
Keywords: Finite element method, Galerkin method, Rayleigh-Ritz method, Hermite interpolation, spline, mollifier, approximation, Sobolev space, boundary value problem
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society