Drift-Randomized Milstein-Galerkin Finite Element Method for Semilinear Stochastic Evolution Equations

TL;DR

This paper rigorously analyzes the strong convergence of a drift-randomized Milstein-Galerkin finite element method for semilinear stochastic evolution equations with multiplicative noise, achieving near first-order temporal convergence without differentiability assumptions.

Contribution

It provides the first strong convergence analysis for the drift-randomized Milstein-Galerkin scheme applied to SEEs with multiplicative noise, establishing near first-order temporal rates.

Findings

Achieves temporal convergence rate of approximately $O(\Delta t)$

Validates theoretical convergence rates through numerical experiments

Extends randomized Galerkin methods to multiplicative noise scenarios

Abstract

Kruse and Wu [Math. Comp. 88 (2019) 2793--2825] proposed a fully discrete randomized Galerkin finite element method for semilinear stochastic evolution equations (SEEs) driven by additive noise and showed that this method attains a temporal strong convergence rate exceeding order $\frac{1}{2}$ without imposing any differentiability assumptions on the drift nonlinearity. They further discussed a potential extension of the randomized method to SEEs with multiplicative noise and introduced the so-called drift-randomized Milstein-Galerkin finite element fully discrete scheme, but without providing a corresponding strong convergence analysis. This paper aims to fill this gap by rigorously analyzing the strong convergence behavior of the drift-randomized Milstein-Galerkin finite element scheme. By avoiding the use of differentiability assumptions on the nonlinear drift term, we establish strong convergence rates in both space and time for the proposed method. The obtained temporal convergence rate is $O(\Delta t^{1-\varepsilon_0})$, where $\Delta t$ denotes the time step size and $\varepsilon_0$ is an arbitrarily small positive number. Numerical experiments are reported to validate the theoretical findings.