On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients

Abstract

In this paper, we introduce a randomized version of the backward Euler method that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we consider Carathéodory-type functions satisfying a one-sided Lipschitz condition. After investigating the well-posedness and the stability properties of the randomized scheme, we prove the convergence to the exact solution with a rate of 0.5 in the root-mean-square norm assuming only that the coefficient function is square integrable with respect to the temporal parameter. These results are then extended to the approximation of infinite-dimensional evolution equations under monotonicity and Lipschitz conditions. Here, we consider a combination of the randomized backward Euler scheme with a Galerkin finite element method. We obtain error estimates that correspond to the regularity of the exact solution. The practicability of the randomized scheme is also illustrated through several numerical experiments.

Publication
Foundations of Computational Mathematics, vol. 19, 1387-1430
Raphael Kruse
Raphael Kruse
Professor

Prof. Dr. Raphael Kruse is the head of the working group “Numerik stochastischer Differentialgleichungen” at Martin-Luther-University Halle-Wittenberg. His research interests include numerical methods and stochastic analysis for stochastic evolution equations and Monte Carlo methods.