Abstract
Mechanisms are given which explain, in a quantitative fashion, "pollution" from nonsmooth regions of a function approximated into smooth regions. In spline L2-projection pollution is negligible. In Fourier series it is governed by a global modulus of continuity in L1 and in Legendre series by a weighted L1 global modulus of continuity which accounts for certain boundary effects.
The three cases considered may serve as models for similar effects in approximation of derivatives in the "h-finite element", the "spectral" and the "p-finite element" methods of solving elliptic partial differential equations. However, the case of the "h-method", which is at present reasonably understood also in two space dimensions, shows that the analogy is far from perfect. One notes, though, that in some more complicated situations with the "h-method" the "pollution" effects are never better than in the simple model considered here. Thus, our rather pessimistic results concerning Fourier series and Legendre series may be of some value in delineating the performances of the "spectral" and "p-finite element" methods.
Examples are given that demonstrate sharpness of the "pollution" mechanisms described. Also, examples of de la Vallée-Poussin type summation methods which reduce "pollution" are considered.
This work was supported by the National Science Foundation, USA.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S.A. Agahanov and G.I. Natanson; The Lebesgue function in Fourier-Jacobi sums, Vestnik Leningrad University, Mathematics, 1968, No. 1, 11–23.
I. Babuska; Lecture at the Finite Element Circus, in Knoxville, Tennessee, October 1983.
I. Babuska, B.A. Szabo and I.N. Katz; The p-version of the finite element method, SIAM J. Numer. Anal. 18, 1981, 515–545.
N.K. Bary; A Treatise on Trigonometric Series, Volume 2, Macmillan, New York, 1964.
J. Descloux; On finite element matrices, SIAM J. Numer. Anal. 9, 1972, 260–265.
J. Douglas Jr., T. Dupont and L.B. Wahlbin; Optimal L∞ error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comp. 29, 1975, 475–483.
L. Fejér; Über die Laplacesche Reihe, Math. Annal. 67, 1909, 76–109.
D. Gottlieb and S.A. Orszag; Numerical Analysis of Spectral Methods: Theory and Applications, SIAM Regional Conference Series in Applied Mathematics, no. 26, SIAM, Philadelphia, Pennsylvania, 1980.
T.H. Gronwall; Über die Laplacesche Reihe, Math. Annal. 74, 1913, 213–270.
G.G. Lorentz; Approximation of Functions, Holt, Rinehart and Winston, New York, New York, 1966.
J.J. Moreau; Approximation en graphe d'une évolution discontinue, RAIRO, Analyse Numerique 12, 1978, 75–84.
S.M. Nikol'skii; On the best approximation of functions satisfying a Lipschitz's condition by polynomials, Izvestia Akad. Nauk. SSSR, Ser. Mat. 10, 1946, 295–322.
J. Nitsche and A.H. Schatz; On local approximation properties of L2 projection on spline subspaces, Applicable Anal. 2, 1972, 161–168.
M.E. Noble; Coefficient properties of Fourier series with a gap condition, Math. Annal. 128, 1954, 55–62 (Correction: 256).
M.J.D. Powell; Approximation Theory and Methods, Cambridge University Press, Cambridge, 1981.
B. Riemann; Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe (1854), Collected Works, 2nd Ed., Leipzig, 1892, 227–271.
A.H. Schatz and L.B. Wahlbin; Interior maximum norm estimates for finite element methods, Math. Comp. 31, 1977, 414–442.
A.H. Schatz and L.B. Wahlbin; On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. Comp. 40, 1983, 47–89.
B. Sendov; Some questions of the theory of approximations of functions and sets in the Hausdorff metric, Russian Math. Surveys 24, 1969, 143–183.
P.K. Suetin; Classical Orthonormal Expansions, Nauka, Moscow, 1979.
G. Szegö; Orthogonal Polynomials, 4th Ed., AMS, Providence, Rhode Island, 1975.
A.F. Timan; A strengthening of Jackson's theorem on the best approximation of continuous functions by polynomials on a finite interval of the real axis, Doklady Akad. Nauk. SSSR, 78, 1951, 17–20.
L.B. Wahlbin; On the sharpness of certain local estimates for \(\mathop {{H^1}}\limits^o\) projections into finite element spaces: Influence of a reentrant corner, Math. Comp. January 1984, to appear.
A. Zygmund; Trigonometric Series, 2nd Ed., (Reprinted), Cambridge University Press, Cambridge, 1968.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1985 Springer-Verlag
About this paper
Cite this paper
Wahlbin, L.B. (1985). A comparison of the local behavior of spline L2-projections, fourier series and legendre series. In: Grisvard, P., Wendland, W.L., Whiteman, J.R. (eds) Singularities and Constructive Methods for Their Treatment. Lecture Notes in Mathematics, vol 1121. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076279
Download citation
DOI: https://doi.org/10.1007/BFb0076279
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-15219-4
Online ISBN: 978-3-540-39377-1
eBook Packages: Springer Book Archive