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

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