ABSTRACT
Definition 7.1.1. A bounded family of operators Th, from to M(−r, h), r>0, is called stable, if there exist δ>0 and C>0 such that for any and all h, |h|<δ0, the following estimate holds:
(7.1.1)
where Ph is the operator of restriction to Ω1h, and C is a constant independent of h=(h1,…, hn). Definition 7.1.2. Functions , are said to converge in
to a function , if
Definition 7.1.3. Functions , are said to converge weakly in to a function , if
Proposition 7.1.1. Suppose that a stable family of operators Th: M(r, h)→M(−r, h) weakly approximates a bounded operator and the equation Tu=f admits a solution for any . Then
(i) the equation Tu=f can have only one solution; (ii) the solutions of the equation Tha(x, h)=f(x, h), where
converge in M(r, h) to the solution of the equation Tu=f.
