Numerical approximation of nonlinear fourth-order SPDEs with additive space-time white noise

TL;DR

This work analyzes the strong numerical approximation of a nonlinear fourth-order SPDE on the 2D torus driven by additive space-time white noise, written as ${\\partial_t u = -δ Δ^2 u - \\mathbf{G}(u) + \\sigma \\xi}$. It introduces a full discretisation using a spectral Galerkin method in space and explicit Euler in time, producing the scheme ${u^{N,n}}$, and proves a near-optimal convergence rate in $L^p$ for the error against the mild solution $v$, namely ${\\left(\\mathbb{E} \,\\sup_{t\\in[0,T]} \\|u^{N,n}(t)-v(t)\\|_{L^2}^p\\right)^{1/p} \\le C (N^{-1+\\varepsilon} + n^{-1+\\varepsilon})}$ for any small $\\varepsilon>0$. The analysis hinges on sharp semigroup estimates for the fourth-order operator $-Δ^2$ and the stochastic sewing lemma to balance spatial and temporal regularity of the noise, enabling almost first-order accuracy in both space and time. The results extend the numerical theory for singular SPDEs by integrating probabilistic sewing techniques with analytic smoothing, and they set a groundwork for more general nonlinearities and geometries on the torus. Potential future work includes treating more complex nonlinearities and surface-growth-type models within this framework.

Abstract

We consider the strong numerical approximation for a fourth-order stochastic nonlinear SPDE driven by space-time white noise on $2$-dimensional torus. We consider its full discretisation with a spectral Galerkin scheme in space and Euler scheme in time. We show the convergence with almost spatial rate $1$ and $1$-temporal rate obtained mainly via \it{stochastic sewing} technique.

Theorems & Definitions (19)

• Definition 2.1
• Theorem 2.2
• Lemma 3.1: Stochastic sewing Lemma, DKP
• Lemma 3.2: Grönwall Lemma. DKP
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 4.1
• proof