  In this paper, we consider a semi-linear stochastic strongly damped wave
equation driven by additive Gaussian noise. Following a semigroup framework, we
establish existence, uniqueness and space-time regularity of a mild solution to
such equation. Unlike the usual stochastic wave equation without damping, the
underlying problem with space-time white noise (Q = I) allows for a mild
solution with a positive order of regularity in multiple spatial dimensions.
Further, we analyze a spatio-temporal discretization of the problem, performed
by a standard finite element method in space and a well-known linear implicit
Euler scheme in time. The analysis of the approximation error forces us to
significantly enrich existing error estimates of semidiscrete and fully
discrete finite element methods for the corresponding linear deterministic
equation. The main results show optimal convergence rates in the sense that the
orders of convergence in space and in time coincide with the orders of the
spatial and temporal regularity of the mild solution, respectively. Numerical
examples are finally included to confirm our theoretical findings.
