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We consider a finite element approximation of a general semi-linearstochastic partial differential equation (SPDE) driven by space-timemultiplicative and additive noise. We examine the full weak convergence rate ofthe exponential Euler scheme when the linear operator is self adjoint andprovide preliminaries results toward the full weak convergence rate fornon-self-adjoint linear operator. Key part of the proof does not rely onMalliavin calculus. Depending of the regularity of the noise and the initialsolution, we found that in some cases the rate of weak convergence is twice therate of the strong convergence. Our convergence rate is in agreement with somenumerical results in two dimensions.

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