Weak convergence for a stochastic exponential integrator and finite element discretization of SPDE for multiplicative \& additive noise

TL;DR

This paper analyzes the weak convergence rates of an exponential Euler scheme combined with finite element discretization for semi-linear SPDEs driven by multiplicative and additive noise, highlighting cases where weak convergence exceeds strong convergence.

Contribution

It provides the first detailed weak convergence analysis for this class of SPDEs with both multiplicative and additive noise, including cases with non-self-adjoint operators.

Findings

Weak convergence rate can be twice the strong convergence rate under certain regularity conditions.

Convergence rates align with numerical results in two-dimensional cases.

Preliminary results extend understanding of weak convergence for non-self-adjoint operators.

Abstract

We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weak convergence rate of the exponential Euler scheme when the linear operator is self adjoint and provide preliminaries results toward the full weak convergence rate for non-self-adjoint linear operator. Key part of the proof does not rely on Malliavin calculus. Depending of the regularity of the noise and the initial solution, we found that in some cases the rate of weak convergence is twice the rate of the strong convergence. Our convergence rate is in agreement with some numerical results in two dimensions.