Convergence of finite element approximations for the one-dimensional stochastic Burgers equation with additive trace-class noise

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 \(L^p([0,T] \times \Omega;H_{D}^{\alpha,q})\) for \(p,q\in[2,\infty)\) and \(\alpha\in[-1,0]\), as well as an almost regularity-optimal estimate in \(L^p(\Omega;C([0,T];L^\infty(\mathcal{O})))\). 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 \(\tau^{1/2-\varepsilon}\) in a discrete analogue of \(L^p(\Omega;C([0,T];L^\infty(\mathcal{O})))\), under the condition \(\tau \leqslant h^2\). Numerical experiments are presented to illustrate the theoretical convergence rates.