Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Polynomial approximation of functions in Sobolev spaces

Authors: Todd Dupont and Ridgway Scott
Journal: Math. Comp. 34 (1980), 441-463
MSC: Primary 65D15; Secondary 41A10
MathSciNet review: 559195
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.

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

  • Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
  • R. ARCANGELI & J. L. GOUT, “Sur l’évaluation de l’erreur d’interpolation de Lagrange dans un ouvert de ${{\text {R}}^n}$,” RAIRO Analyse Numérique, v. 10, 1976, pp. 5-27.
  • Alan Berger, Ridgway Scott, and Gilbert Strang, Approximate boundary conditions in the finite element method, Symposia Mathematica, Vol. X (Convegno di Analisi Numerica, INDAM, Rome, 1972) Academic Press, London, 1972, pp. 295–313. MR 0403258
  • J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 263214, DOI
  • J. H. Bramble and S. R. Hilbert, Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math. 16 (1970/71), 362–369. MR 290524, DOI
  • V. I. Burenkov, Sobolev’s integral representation and Taylor’s formula, Trudy Mat. Inst. Steklov. 131 (1974), 33–38, 244 (Russian). Studies in the theory of differentiable functions of several variables and its applications, V. MR 0357711
  • P. G. Ciarlet and C. Wagschal, Multipoint Taylor formulas and applications to the finite element method, Numer. Math. 17 (1971), 84–100. MR 287666, DOI
  • Jim Douglas Jr., Todd Dupont, Peter Percell, and Ridgway Scott, A family of $C^{1}$ finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems, RAIRO Anal. Numér. 13 (1979), no. 3, 227–255 (English, with French summary). MR 543934, DOI
  • Todd Dupont and Ridgway Scott, Constructive polynomial approximation in Sobolev spaces, Recent advances in numerical analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 41, Academic Press, New York-London, 1978, pp. 31–44. MR 519055
  • R. E. EWING, “Alternating direction Galerkin methods for some third and fourth order equations.” (To appear.)
  • Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
  • Pierre Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 207–274. MR 0466912
  • L. HÖRMANDER, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963.
  • Pierre Jamet, Estimation de l’erreur d’interpolation dans un domaine variable et application aux éléments finis quadrilatéraux dégénérés, Méthodes numériques en mathématiques appliquées (Sém. Math. Sup., Quatorzième Session, Univ. Montréal, Montreal, Que., 1975) Les Presses Univ. Montréal, Montreal, Que., 1977, pp. 55–100. Sém. Math. Sup., No. 60 (French). MR 0445863
  • Jean Meinguet, Structure et estimations de coefficients d’erreurs, RAIRO Anal. Numér. 11 (1977), no. 4, 355–368, iv (French, with English summary). MR 474761, DOI
  • Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
  • Ridgway Scott, Interpolated boundary conditions in the finite element method, SIAM J. Numer. Anal. 12 (1975), 404–427. MR 386304, DOI
  • S. L. SOBOLEV, Applications of Functional Analysis in Mathematical Physics, Transl. Math. Monographs, Vol. 7, Amer. Math. Soc., Providence, R. I., 1963.
  • Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • B. L. VAN DER WAERDEN, Modern Algebra II, 2nd ed., Ungar, New York, 1950.
  • A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D15, 41A10

Retrieve articles in all journals with MSC: 65D15, 41A10

Additional Information

Article copyright: © Copyright 1980 American Mathematical Society