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

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.