Stable discretization methods with external approximation schemes (Q1908334)

For a given initial value problem (1) \(Ax = b\), \(x \in X\) and the corresponding discretized problem \(A_n x_n = b_n\), \(x_n \in X_n\) an external approximation scheme of the form \(\Pi_0 = \{X, F, X_n, X_n^*, W, A_n\), \(R_n, K_n, E_n\}\) is considered. Here \(X\), \(F\), \(X_n\) are real Banach spaces, \(X^*\) is the dual of \(X\), \(R_n : X \to X_n,\) \(K_n : X_n \to X_n\), \(E_n : X_n \to F\), \(W : X \to F\), \(A : X \to X^*\) and \(A_n : X_n \to X^*\) are operators satisfying suitable linearity, continuity, injectivity or boundedness properties. When \(K_n = 1_{X_n}\) the scheme \(\Pi_0\) coincides with that of \textit{R. Schumann} and \textit{E. Zeidler} [Numer. Funct. Anal. Optimization 1, 161-194 (1979; Zbl 0463.65068)], while for \(F = X\), \(W = 1_X\), \(K_n = 1_{X_n}\) it reduces to the inner approximation schemes of \textit{W. V. Petryshyn} [Proc. Sympos. Pure Math. 18, Part 1, 206-233 (1970; Zbl 0232.47070); J. Math. Mech. 17, 353-372 (1967; Zbl 0162.20202)]. The main result of the paper (Theorem 2.1) can be briefly stated as follows: If the external approximation scheme \(\Pi_0\) is an admissible one and has consistency and stability, then the equation (1) is uniquely approximation-solvable iff the operator \(A\) is \(A\)-proper. Finally an application of this result in the case \(X\) and \(X_n\) are Sobolev spaces is sketched.