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

Ben Goldys, Agus L. Soenjaya and Thanh TRan

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 problem converges in probability to that of the original problem. A fully discrete numerical scheme is then proposed for the truncated system, and we establish both convergence in probability and strong convergence (with quantitative rates) for the two fields used in the mixed formulation.

Keywords: Fourth order PDE, stochastic PDE, finite element method, strong solution, multiplicative noise.

AMS Subject Classification: Primary 60H15;; secondary 60H35; 65C30; 65M12; 65M60.