By H. H. Schaefer

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**Extra info for Banach Lattices and Positive Operators (Grundlehren Der Mathematischen Wissenschaften Series, Vol 215)**

**Sample text**

N. de G. Allen, Relaxation Methods in Engineering and Science, McGraw-Hili, New York, 1954. I. Babuska, M. Prager and M. 2. 12 See also [Az, p. 203], and E. A. Volkov, Vychisl. , 1 (1957), pp. 34-61 and 62-80. 26 LECTURE 3 Using such considerations, global convergence as h ! 0 was first proved for the Laplace ~E on a square mesh by R. G. D. Richardson in 1917 and by Phillips and Wiener in 1922; the aim of these authors was to establish existence theorems for solutions of the Dirichlet problem for V 2 u = 0 from algebraic existence theorems for V~u = O.

Generalized Networks, MRI Symposium Series 16, Brooklyn Polytechnic Press, New York, 1966, and the references of Lecture 8, footnote 8. j + 1 + O(h 4 ). Instead of using truncated Taylor series to derive difference approximations to derivatives, one can use integral formulas. Careful discussions of this approach may be found in [FW], [KK] , [V] and [W]. In either case, the most useful fact to be deduced from such a priori error estimates is the principle that the error (for a uniform mesh) is typically asymptotic to Mh" + O(h"+ 1) for some positive integer n.

SIAM J. Numer. , 7 (1970), pp. 623-656. 6 G. Birkhoff and 1. B. Diaz, Quart. Appl. , 13 (1956), pp. 432-443; see also G. BirkholI and R. B. Kellogg, Proc. Symp. Generalized Networks, MRI Symposium Series 16, Brooklyn Polytechnic Press, New York, 1966, and the references of Lecture 8, footnote 8. j + 1 + O(h 4 ). Instead of using truncated Taylor series to derive difference approximations to derivatives, one can use integral formulas. Careful discussions of this approach may be found in [FW], [KK] , [V] and [W].