By George Biddell Airy
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N^rsett (1978), Order stars and stability theorems, to appear in BIT. 36. 0. B. Widlund (1967), A note on unconditionally stable linear multistep methods, BIT 7, 65-70. 37. R. Warming and R, Beam (1978), An extension of A-stability to ADI methods. Report at Ames Research Center, NASA, Moffet Field, California. 38. R. Jeltsch (1976), Note on A-stability of multistep multiderivative methods, BIT 16, 74-78. 39. U. Grenander and G. Szegb* (1958), Toeplitz forms and their applica tions. Univ. Calif.
B) and . |a| < m. 3) is seen to hold for all u e *>'(B). 4) that Q It also follows is a Lipschitz continuous mapping m of L^(B) into the Banach space P . m Of interest in the sequel will be the error u - Q u. Since Q^u is a polynomial and hence defined on all of R , n it is natural to ask how w e l l Q u agrees with u on any domain ra containing B where u may be defined. set in R n Let 0 be a bounded open that is star-shaped with respect to every point in CO B. ) (x-y) k(x,y) a and k(x,y) s J 0 cp(x+s s (y-x))ds.
Perturbation from nodes for a straight-edged triangle, w e conclude K £ 7 is sufficient for h small. W e can estimate the error u - ^ u as follows. Let T e T . 2) . 2 implies ||u-J u|| < Ch(T) l/(T) lul n 1 that _ m ' . 1 where ^ C * (tn/V^I) ( S |p|=m ) p" (l+K). 2 Squaring and summing over TeT, w e find ||u-J ull T 0 * Ch(T) m lul^ , where the norms refer to t h e domain wise quadratic interpolation ||u-J u|| T 0 * 10p" 2 h(T) 3 In the case of piece- (m=3), w e thus have |u| . 3 T O D D D U P O N T A N D RIDGWAY SCOTT 44 Remarks.
A treatise on trigonometry by George Biddell Airy