Weak order for the discretization of the stochastic heat equation
Provides theoretical weak convergence rates for SPDE discretization under relaxed assumptions on the covariance operator, benefiting numerical analysts and practitioners in stochastic PDEs.
The paper proves a weak error bound of O(h^{2γ} + Δt^γ) for finite element and implicit Euler discretization of the stochastic heat equation, where the convergence rate is twice that of pathwise approximations.
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.