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

TL;DR

This paper introduces a randomized backward Euler method for nonlinear evolution equations with irregular coefficients, demonstrating convergence and stability, and extending to infinite-dimensional cases with practical numerical experiments.

Contribution

It presents a novel randomized backward Euler scheme applicable to irregular coefficients and extends analysis to infinite-dimensional evolution equations with error estimates.

Findings

Converges with rate 0.5 in root-mean-square norm.

Applicable to infinite-dimensional evolution equations.

Validated by numerical experiments.

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 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.