Unconditional convergence of DIRK schemes applied to dissipative evolution equations

Abstract

In this paper we prove the convergence of algebraically stable DIRK schemes applied to dissipative evolution equations on Hilbert spaces. The convergence analysis is unconditional as we do not impose any restrictions on the initial value or assume any extra regularity of the solution. The analysis is based on the observation that the schemes are linear combinations of the Yosida approximation, which enables the usage of an abstract approximation result for dissipative maps. The analysis is also extended to the case where the dissipative vector field is perturbed by a locally Lipschitz continuous map. The efficiency and robustness of these schemes are finally illustrated by applying them to a nonlinear diffusion equation.