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 $$ dX_t+AX_t dt = Q^{1/2} d W_t, \quad X_0=x \in H, \quad t\in[0,T], $$ driven by a Gaussian space time noise whose covariance operator $Q$ is given. We assume that $A^{-α}$ is a finite trace operator for some $α>0$ and that $Q$ is bounded from $H$ into $D(A^β)$ for some $β\geq 0$. 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 $Δt=T/N$). We define a discrete solution $X^n_h$ and for suitable functions $ϕ$ defined on $H$, we show that $$ |\E ϕ(X^N_h) - \E ϕ(X_T) | = O(h^{2γ} + Δt^γ) $$ \noindent where $γ<1- α+ β$. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations.