Discrete forms of Friedrichs ’ inequalities in the finite element method

Auxihary theorems allowing to extend the theory of curved finite éléments introduced m [2], [3] to the case ofboundary value problems with varions stable and unstable boundary conditions are proved As an example the problem of bending of thin elastic plates is considered Résumé — OM démontre des théorèmes auxiliaires qui permettent d'étendre la théorie des éléments finis courbes présentée dans [2], [3] au cas de problèmes aux limites avec des conditions aux limites stables et instables diverses Comme exemple, on présente le problème de la flexion élastique des plaques minces Let Qh be a finite element approximation of a given domain Q and W\ a finite element subspace of the space C~(Qh) of all fonctions definçd on Q^ which have continuous derivatives up to order s — 1 on Qh {s ̂ 1). The main aim of this paper is to show that for h < K (where W is sufficiently small) the constants K(Q.h) appeanng in Friedrichs' inequality and related inequalities written for fonctions from Wh can be substituted by constants independent on k This resuit allows to extend the theory of curved finite éléments developed by Ciarlet and Raviart [2] and Ciarlet [3] to the case of boundary value problems with various stable and unstable boundary conditions. The inequahties appeanng in this paper are called discrete forms of Friedrichs' inequahties because they are wntten only for the fonctions from the finite dimensional spaces Wh. As usual, the symbol H (Q) will dénote the Sobolev space H\Q) = { v e L2(Q) : D*v G L2(Q) V | a | < k } (*) Reçu le 5 mai 1980 C) Computing Center of the Techmcal Umversity, Obrâncû miru 21, 60200 Brno, Czechoslovakia R A ï R O Analyse numérique/Numerical Analysis, 0399-0516/1981/265/S 5 00 © Bordas-Dunod