  A numerical analysis for the fully discrete approximation of an operator
Lyapunov equation related to linear SPDEs (stochastic partial differential
equations) driven by multiplicative noise is considered. The discretization of
the Lyapunov equation in space is given by finite elements and in time by a
semiimplicit Euler scheme. The main result is the derivation of the rate of
convergence in operator norm. Moreover, it is shown that the solution of the
equation provides a representation of a quadratic and path dependent functional
of the SPDE solution. This fact yields a deterministic numerical method to
compute such functionals. As a secondary result, weak error rates are
established for a fully discrete finite element approximation of the SPDE with
respect to this functional. This is obtained as a consequence of the
approximation analysis of the Lyapunov equation. It is the first weak
convergence analysis for fully discrete finite element approximations of SPDEs
driven by multiplicative noise that obtains double the strong rate of
convergence, especially for path dependent functionals and smooth spatial
noise. Numerical experiments illustrate the results empirically and it is
demonstrated that the deterministic method has advantages over Monte Carlo
sampling in a stability context.
