On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients
Preprint, 2017

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 Carath\'eodory 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 numerical solution 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.

ordi- nary differential equations

evolution equations

backward Euler method

Monte Carlo method

Galerkin finite element method.

stratified sampling

Författare

Monika Eisenmann

Organisation okänd

Mihaly Kovacs

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Raphael Kruse

Organisation okänd

Stig Larsson

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Ämneskategorier

Beräkningsmatematik

Sannolikhetsteori och statistik

Fundament

Grundläggande vetenskaper

Mer information

Skapat

2018-09-29