Numerische Mathematik

, Volume 21, Issue 3, pp 270–278 | Cite as

Superconvergence for galerkin methods for the two point boundary problem via local projections

  • Jim DouglasJr.
  • Todd Dupont


The two point boundary problemy′'-a(x)y−b(x)y=-f(x), o<x<1,y(0)=y(1)=0, is first solved approximately by the standard Galerkin method, (Y′, υ′) + (aY′+bY, υ)=(f, υ), υ ∈ ℳ 1 0 (r, δ), for a function Y∈ℳ 1 0 (r, δ), the space ofC1-piecewise-ψ-degree-polynomials vanishing atx=0 andx=1 and having knots at {x 0 ,x 1 , ...,x M }=δ. ThenY is projected locally into a polynomial of higher degree by means of one of several projections. It is then shown that higher-order convergence results locally, provided thaty is locally smooth and δ is quasi-uniform.


Mathematical Method Point Boundary Boundary Problem Galerkin Method Convergence Result 
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  1. 1.
    Boor, C., de, Swartz, B.: Collocation at Gaussian points. To appearGoogle Scholar
  2. 2.
    Bramble, J. H., Schatz, A. H.: Rayleigh-Ritz-Galerkin methods for Dirichlet's problem using subspaces without boundary conditions. Comm. Pure Appl. Math.23, 653–675 (1970)Google Scholar
  3. 3.
    Ciarlet, P. G., Schultz, M. H., Varga, R. S.: Numerical methods of high-order accuracy for nonlinear boundary value problems, I. Numer. Math.9, 394–430 (1967)Google Scholar
  4. 4.
    Davis, P. J.: Interpolation and approximation. New York: Blaisdell 1963Google Scholar
  5. 5.
    Douglas, J., Jr., Dupont, T.: Some superconvergence results for Galerkin methods for the approximate solution of two point boundary problems, to appear in the Proceedings of the Conference on Numerical Analysis, Royal Irish Academy, Dublin, 1972Google Scholar
  6. 6.
    Douglas, J., Jr., Dupont, T.: A superconvergence result for the approximate solution of the heat equation by a collocation method. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz (ed.). New York: Academic Press 1972Google Scholar
  7. 7.
    Douglas, J., Jr., Dupont, T.: The approximate solution of nonlinear parabolic equations by collocation methods usingC 1-piecewise-polynomial spaces, to appearGoogle Scholar
  8. 8.
    Nitsche, J.: Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens. Numer. Math.11, 346–348 (1968)Google Scholar
  9. 9.
    Thomée, V.: Spline approximation and difference schemes for the heat equation (same book as in 6 above)Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • Jim DouglasJr.
    • 1
  • Todd Dupont
    • 1
  1. 1.University of ChicagoChicagoUSA

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