Weak approximation of stochastic partial differential equations: the nonlinear case

We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that, as is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case of a semilinear stochastic heat equation driven by a space-time white noise.