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 $$dX_t+AX_t dt = Q^{1/2} d W_t, \quad X_0=x \in H, \quad t\in[0,T],$$ 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 implicit Euler schemes in time (parameter $\Delta t=T/N$). We define a discrete solution $X^n_h$ and for suitable functions $\phi$ defined on $H$, we show that $$|\E \phi(X^N_h) - \E \phi(X_T) | = O(h^{2\gamma} + \Delta t^\gamma)$$ \noindent 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.