  The recently proposed statistical finite element (statFEM) approach
synthesises measurement data with finite element models and allows for making
predictions about the true system response. We provide a probabilistic error
analysis for a prototypical statFEM setup based on a Gaussian process prior
under the assumption that the noisy measurement data are generated by a
deterministic true system response function that satisfies a second-order
elliptic partial differential equation for an unknown true source term. In
certain cases, properties such as the smoothness of the source term may be
misspecified by the Gaussian process model. The error estimates we derive are
for the expectation with respect to the measurement noise of the $L^2$-norm of
the difference between the true system response and the mean of the statFEM
posterior. The estimates imply polynomial rates of convergence in the numbers
of measurement points and finite element basis functions and depend on the
Sobolev smoothness of the true source term and the Gaussian process model. A
numerical example for Poisson's equation is used to illustrate these
theoretical results.
