On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients
Artikel i vetenskaplig tidskrift, 2019

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.

Ordinary differential equations

Backward Euler method

Evolutions equations

Monte Carlo method

Garlekin finite element method


Monika Eisenmann

Technische Universität Berlin

Mihaly Kovacs

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Raphael Kruse

Technische Universität Berlin

Stig Larsson

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Foundations of Computational Mathematics

1615-3375 (ISSN) 1615-3383 (eISSN)

Vol. 19 6 1387-1430



Sannolikhetsteori och statistik

Matematisk analys


Grundläggande vetenskaper



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