By S. Albertoni (auth.), Jacques Louis Lions (eds.)

S. Albertoni: Alcuni metodi di calcolo nella teoria della diffusione dei neutroni.- I. Babuska: Optimization and numerical balance in computations.- J.H. Bramble: blunders estimates in elliptic boundary worth problems.- G. Capriz: The numerical method of hydrodynamic problems.- A. Dou: strength inequalities in an elastic cylinder.- T. Doupont: at the lifestyles of an iterative strategy for the answer of elliptic distinction equation with a stronger paintings estimate.- J. Douglas, J.R. Cannon: The approximation of harmonic and parabolic capabilities of half-spaces from inside data.- B.E. Hubbard: errors estimates within the mounted Membrane problem.- ok. Jorgens: Calculation of the spectrum of a Schrödinger operator.- A. Lasota: Contingent equations and boundary price problems.- J.L. Lions: Réduction � des problèmes du kind Cauchy-Kowalewska.- J.L. Lions: Problèmes aux limites non homogènes � données irrégulières; une méthode d’approximation.- J.L. Lions: Remarques sur l’approximation régularisée de problèmes aux limites.- W.V. Petryshyn: at the approximation-solvability of nonlinear useful equations in normed linear spaces.- P.A. Raviart: Approximation des équations d’évolution par des méthodes variationnelles.- M. Sibony, H. Brezis: Méthodes d’approximation et d’itération pour les operateurs monotones.- V. Thomee: a few themes in balance conception for partial distinction operators.

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25), given more exactly and in detail more in (25). As an illustration I shall give the following example: Example 2 . 3 . 2 11 sin x f(x) dn for ffx) = e e = 10,50. 3 we s e e the e r r o r when using the formulas Compute also J(f) = Let us assume O(= 1 . 1 - ' ,P . e. on n resp. n ( l t C ( ) . We T and T; 2) In n s e e that the computation without the sue of derivatives i s more advantageous. This agrees fully with the theoretic investigations. In accordance with the theor e m the e r r o r of the formula with derivatives is nearly three times l a r g e r than that of the formula without derivatives.

The previously mentioned p r o c e s s e s a r e a ) B1 solution, I b),b) B c) solutinn, 1 Bo solution. I shall now show the meaning of t h i s t h e o r e m by m e a n s of the following example : Example 4. 1. We s h a l l with the standard solve the initial Runge-Kutta method problem f o r the equation of the 4th d e g r e e . 1 . 1 y(a t -) 1s t o be solved. H e r e we obviously have C = - . Our t a s k is 2 2 to e s t i m a t e 1 y (n) - y (n) 1 independence on n. Since t h e solution of n n - ( 4 .

F r o m this table we see that a n optimal formula used in a n n inconvenient s p a c e may give bad r e s u l t s . We s e e that the conclusion of the convenience of the optimal formula i s v e r y llunstablell with r e s p e c t to the choic e of H) . 2 F o r e v e r y periodic s p a c e H A (n. 11) This theorem periodic shows that the trapezoid formula cannot space. Nevertheless t h i s formula i s v e r y advantageous in tice. 3 . Let H be a b e optimal in a periodic in the following for all periodic s p a c e s (except statement: space.