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

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 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.

37 pages, 3 figures

References in corpus (2)

• Error analysis of randomized Runge-Kutta methods for differential equations with time-irregular coefficients
• Optimal Error Estimates for Fully Discrete Galerkin Approximations of Semilinear Parabolic Equations

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