Convergence of finite element approximations for the one-dimensional stochastic Burgers equation with additive trace-class noise
Year: 2026
Type: preprint
Abstract: This paper investigates finite element approximations of the one-dimensional viscous stochastic Burgers equation with additive trace-class noise. For the $P_2$ finite element spatial semi-discretization, we derive strong error estimates that are optimal with respect to regularity in \(L^p([0,T] \times Ω;H_{D}^{α,q})\) for \(p,q\in[2,\infty)\) and \(α\in[-1,0]\), as well as an almost regularity-optimal estimate in \(L^p(Ω;C([0,T];L^\infty(\mathcal{O})))\). Furthermore, we derive weak error estimates for moments of both terminal $L^q$-norms and space-time $L^p(0,T;L^q(\mathcal{O}))$-norms, with weak convergence rates (nearly) twice the corresponding strong ones. For the fully discrete scheme, which combines the \(P_2\) finite element method in space with a drift-implicit Euler--Maruyama scheme in time, we establish a strong temporal convergence rate of order \(τ^{1/2-\varepsilon}\) in a discrete analogue of \(L^p(Ω;C([0,T];L^\infty(\mathcal{O})))\), under the condition \(τ\leqslant h^2\). Numerical experiments are presented to illustrate the theoretical convergence rates.
Source: arXiv (Cornell University)
Authors: Li, Binjie, Xie, Xiaoping, Zhou, Qin
Institutions: Sichuan University, China West Normal University
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Awards: G1795423594, G3413728038
OpenAlex ID: W7212288167