On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients
Eisenmann, Monika LU
; Kovács, Mihály
; Kruse, Raphael
and Larsson, Stig
(2019)
In Foundations of Computational Mathematics
19.
p.1387-1430
- 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... (More)
- 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. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/b453887b-663c-4506-af15-66771207b68a
- author
- Eisenmann, Monika
LU
; Kovács, Mihály
; Kruse, Raphael
and Larsson, Stig
- publishing date
- 2019
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Foundations of Computational Mathematics
- volume
- 19
- pages
- 1387 - 1430
- external identifiers
-
- scopus:85059595718
- DOI
- 10.1007/s10208-018-09412-w
- language
- English
- LU publication?
- no
- id
- b453887b-663c-4506-af15-66771207b68a
- date added to LUP
- 2024-10-07 15:03:06
- date last changed
- 2025-04-04 15:00:56
@article{b453887b-663c-4506-af15-66771207b68a,
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.}},
author = {{Eisenmann, Monika and Kovács, Mihály and Kruse, Raphael and Larsson, Stig}},
language = {{eng}},
pages = {{1387--1430}},
series = {{Foundations of Computational Mathematics}},
title = {{On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients}},
url = {{http://dx.doi.org/10.1007/s10208-018-09412-w}},
doi = {{10.1007/s10208-018-09412-w}},
volume = {{19}},
year = {{2019}},
}