A functional calculus approach for the rational approximation with nonuniform partitions (Q950620)

The authors generalize some results of \textit{P. Brenner} and \textit{V. Thomée} [SIAM J. Numer. Anal. 16, 683--694 (1979; Zbl 0413.41011)] on the stability and convergence of rational approximation schemes of bounded semigroups for uniform time steps. The purpose of the paper is to approximate \( e^{tA}\) by suitable operators \(P_{n}(A)\) by using nonuniform partitions \(\tau_{1},...,\tau_{n}\) of the time interval [0,t]. If \(R\) is a rational function \[ P_{n}(A)=R(\tau_{1}A)... R(\tau_{n}A). \] The choice of rational functions is motivated by the fact that they appear in such important schemes as backward Euler and Crank-Nicolson. The paper is organized as follows: Notation and main assumptions; \textit{M}-functional calculus; On the rate of convergence; Sharpness of the Convergence Theorem; Approximation of the derivative; References(12).