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On a Randomized Backward Euler Method for Nonlinear Evolution Equations with Time-Irregular Coefficients

Journal article
Authors Monika Eisenmann
Mihaly Kovacs
Raphael Kruse
Stig Larsson
Published in Foundations of Computational Mathematics
Volume 19
Issue 6
Pages 1387–1430
ISSN 1615-3375
Publication year 2019
Published at Department of Mathematical Sciences
Pages 1387–1430
Language en
Keywords Backward Euler method, Evolution equations, Galerkin finite element method, Monte Carlo method, Ordinary differential equations
Subject categories Mathematics


© 2018, SFoCM. 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.

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