Estimates of variable stepsize Runge-Kutta methods for sectorial evolution equations with nonsmooth data
We consider variable stepsize Runge-Kutta methods for semilinear evolution equations with a sectorial operator in the linear part. For nonsmooth initial data error estimates are derived that show the interplay of weak singularities and the classical order of convergence. There are no uniformity assumptions on the stepsizes, but we assume a Lipschitz condition for the nonlinearity and a stability function for the method that is less than I on the critical sector and vanishes at infinity. Using an extended operational calculus the proof combines a rearrangement trick with a discrete Gronwall estimate including weak singularities. Our main theorem complements respectively extends well-known results of Bakaev, Gonzalez, Lubich, Ostermann and Palencia. (C) 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.
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