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We consider the numerical approximation of the mild solution to a semilinearstochastic wave equation driven by additive noise. For the spatialapproximation we consider a standard finite element method and for the temporalapproximation, a rational approximation of the exponential function. We firstshow strong convergence of this approximation in both positive and negativeorder norms. With the help of Malliavin calculus techniques this result is thenused to deduce weak convergence rates for the class of twice continuouslydifferentiable test functions with polynomially bounded derivatives. Underappropriate assumptions on the parameters of the equation, the weak rate isfound to be essentially twice the strong rate. This extends earlier work by oneof the authors to the semilinear setting. Numerical simulations illustrate thetheoretical results.

Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise