  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.
