Erratum to: On the Crank-Nicolson scheme once again
Let \((\tau_j)_{j\in\mathbb{N}}\) be a sequence of strictly positive real numbers, and let \(A\) be the generator of a bounded analytic semigroup in a Banach space \(X\). Put \(A_n=\prod_{j=1}^n(I+\frac{1}{2}\tau_jA)(I-\frac{1}{2}\tau_jA)^{-1}\), and let \(x\in X\). Define the sequence \((x_n)_{n\in\mathbb{N}}\subset X\) by the Crank-Nicolson scheme: \(x_n = A_n x\). In a previous paper [``On the Crank-Nicolson scheme once again'', J. Evol. Equ. 11, No. 2, 457--476 (2011; Zbl 1239.47035)], the author made a serious flaw that affects the main stability result. In this Erratum, the author gives the correct formulation and proves that the Crank-Nicolson scheme is stable in the sense that \(\sup_{n\in\mathbb{N}}\| A_nx\| < \infty\), provided that a suitable inequality holds.
scientific article; zbMATH DE number 5997673