A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations

TL;DR

This paper introduces a novel fully discrete numerical method combining Galerkin finite elements with a randomized Runge-Kutta scheme for solving semilinear stochastic evolution equations driven by additive noise, proving convergence without differentiability constraints.

Contribution

The paper presents a new fully discrete scheme that merges Galerkin finite elements with randomized Runge-Kutta methods, achieving optimal convergence rates without requiring nonlinearity differentiability.

Findings

Convergence proven in $L^p$-norm for the proposed method.

Achieves the same temporal order as Milstein-Galerkin methods.

Extension to spectral approximation of Wiener process included.

Abstract

In this paper the numerical solution of non-autonomous semilinear stochastic evolution equations driven by an additive Wiener noise is investigated. We introduce a novel fully discrete numerical approximation that combines a standard Galerkin finite element method with a randomized Runge-Kutta scheme. Convergence of the method to the mild solution is proven with respect to the $L^p$-norm, $p \in [2,\infty)$. We obtain the same temporal order of convergence as for Milstein-Galerkin finite element methods but without imposing any differentiability condition on the nonlinearity. The results are extended to also incorporate a spectral approximation of the driving Wiener process. An application to a stochastic partial differential equation is discussed and illustrated through a numerical experiment.