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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


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).


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.

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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.

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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).

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