Drift-Randomized Milstein-Galerkin Finite Element Method for Semilinear Stochastic Evolution Equations
Xiao Qi, Yue Wu, Yubin Yan
TL;DR
This work addresses numerical approximation of semilinear stochastic evolution equations with multiplicative noise by introducing the drift-randomized Milstein–Galerkin finite element (DRMGFE) method. The method couples a drift-randomized Milstein time discretization with a Galerkin finite element spatial discretization, achieving strong convergence without differentiability assumptions on the drift term $F$. The main result proves an error bound of the form $O(Δt^{1-ε_0} + h^{2-ε_0})$, supported by a discrete martingale inequality and comprehensive a priori estimates; numerical experiments in 1D and 2D corroborate the theoretical rates. This advances high-order, stable SPDE solvers for problems with multiplicative noise and practical regularity assumptions, with demonstrated robustness and efficiency.
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 $O(Δt^{1-\varepsilon_0})$, where $Δt$ denotes the time step size and $\varepsilon_0$ is an arbitrarily small positive number. Numerical experiments are reported to validate the theoretical findings.