Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The construction of a priori bounds for the solution of a two point boundary value problem with finite elements I

Die Konstruktion von a-priori-Schranken zur Lösung eines Randwertproblems mit finiten Elementen I

Abstract

Let û be the solution of a boundary value problem for an ordinary differential equation of the second order. Function boundsv andw are constructed to û such thatv ≦ û ≦w. From this other bounds are derived for the derivatives û′ and û″. To this end a collocation method with finite elements is used. The inclusion property is proven with the aid of theorems on differential inequalities. Leth be the maximal step size and letk be an arbitrary natural number. Then the accuracy can be made to have arbitrarily high order such thatw−v=C(h 2k).

Zusammenfassung

Es sei û die Lösung eines Randwertproblems mit einer gewöhnlichen Differentialgleichung zweiter Ordnung. Zu û werden Schrankenfunktionenv undw konstruiert mitv ≦ û ≦w, daraus folgen ähnliche Aussagen für die Ableitungen û′ und û″. Es wird ein Kollokationsverfahren mit finiten Elementen benutzt; die Schrankeneigenschaften werden durch Differential-Ungleichungs-Sätze gesichert. Isth die maximale Schrittweite undk eine beliebige natürliche Zahl, so lassen sich beliebig hohe Genauigkeitsordnungenw−v=O(h 2k) erzielen.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    Keller, H. B.: Numerical methods for two-point boundary-value problems. Blaisdell 1968.

  2. [2]

    Küpper, T.: Einschließungsaussagen bei Differentialoperatoren zweiter Ordnung durch punkt-weise Ungleichungen. Num. Math.30, 93–101 (1978).

  3. [3]

    Moore, R. E.: Interval analysis. Prentice-Hall 1966.

  4. [4]

    Prenter, P. M.: Splines and variational methods. Wiley-Interscience 1975.

  5. [5]

    Varga, R. S.: Matrix iterative analysis. Prentice-Hall 1962.

  6. [6]

    Walter, W.: Differential and integral inequalities. Springer 1970.

Download references

Author information

Additional information

This paper has been written while the author was a visiting professor at the Computer Science Department of the University of Calgary, Canada, from July to September of 1978. I wish to thank all of them who helped me and made my stay possible, especially professor Jon Rokne of this department.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Nickel, K. The construction of a priori bounds for the solution of a two point boundary value problem with finite elements I. Computing 23, 247–265 (1979). https://doi.org/10.1007/BF02252131

Download citation

Keywords

  • Differential Equation
  • Ordinary Differential Equation
  • Computational Mathematic
  • Natural Number
  • Point Boundary