Weak order for the discretization of the stochastic heat equation

In this paper we study the approximation of the distribution of $X_t$ Hilbert-valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as

$$\mathrm{d} X_t+AX_t \, \mathrm{d} t = Q^{1/2} \mathrm{d} W(t), \quad X_0=x \in H, \quad t\in[0,T],$$

driven by a Gaussian space time noise whose covariance operator $Q$ is given. We assume that $A^{-\alpha}$ is a finite trace operator for some $\alpha>0$ and that $Q$ is bounded from $H$ into $D(A^\beta)$ for some $\beta\geq 0$. It is not required to be nuclear or to commute with $A$.

The discretization is achieved thanks to finite element methods in space (parameter $h>0$) and a $\theta$-method in time (parameter $\Delta t=T/N$). We define a discrete solution $X^n_h$ and for suitable functions $\varphi$ defined on $H$, we show that

$$\vert\mathbb{E} \, \varphi(X^N_h) - \mathbb{E} \, \varphi(X_T) \vert = O(h^{2\gamma} + \Delta t^\gamma)$$

where $\gamma<1- \alpha + \beta$. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations.