Weak order for the discretization of the stochastic heat equation

Abstract

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 $$d{X}_t+A{X}_{t}dt=Q^{1/2}d{W}_t,X_0=x\in H,t\in [0,T],$$ 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 $\beta \ge 0$. 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 $X_h^n$ and for suitable functions $\varphi$ defined on $H$, we show that $$ |\E ϕ(X^N_h) - \E ϕ(X_T) | = O(h^{2γ} + Δt^γ) $$ \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.