A mixed finite element method for a class of fourth-order stochastic evolution equations with multiplicative noise

TL;DR

This paper introduces a novel mixed finite element method for approximating solutions to complex fourth-order stochastic PDEs with multiplicative noise, ensuring convergence and applicability to various physical models.

Contribution

It develops a fully discrete semi-implicit scheme using a truncate-then-discretise approach for challenging nonlinear stochastic equations, with proven convergence and error rates.

Findings

The scheme converges strongly with quantitative rates.

Numerical simulations confirm theoretical convergence.

Applicable to multiple physically relevant stochastic models.

Abstract

We develop a fully discrete, semi-implicit mixed finite element method for approximating solutions to a class of fourth-order stochastic partial differential equations (SPDEs) with non-globally Lipschitz and non-monotone nonlinearities, perturbed by spatially smooth multiplicative Gaussian noise. The proposed scheme is applicable to a range of physically relevant nonlinear models, including the stochastic Landau--Lifshitz--Baryakhtar (sLLBar) equation, the stochastic convective Cahn--Hilliard equation with mass source, and the stochastic regularised Landau--Lifshitz--Bloch (sLLB) equation, among others. To overcome the difficulties posed by the interplay between the nonlinearities and the stochastic forcing, we adopt a `truncate-then-discretise' strategy: the nonlinear term is first truncated before discretising the resulting modified problem. We show that the strong solution to the truncated system converges in probability to that of the original problem. A fully discrete numerical scheme is then proposed for the truncated problem. Assuming initial data in $\mathbb{H}^2$, we utilise parabolic smoothing estimates and the temporal H\"older continuity of the solution to establish both convergence in probability and strong convergence (with quantitative rates) for the two fields used in the mixed formulation. Numerical simulations are provided to support the theoretical results.