  We consider an elliptic linear-quadratic parameter estimation problem with a
finite number of parameters. A novel a priori bound for the parameter error is
proved and, based on this bound, an adaptive finite element method driven by an
a posteriori error estimator is presented. Unlike prior results in the
literature, our estimator, which is composed of standard energy error residual
estimators for the state equation and suitable co-state problems, reflects the
faster convergence of the parameter error compared to the (co)-state variables.
We show optimal convergence rates of our method; in particular and unlike prior
works, we prove that the estimator decreases with a rate that is the sum of the
best approximation rates of the state and co-state variables. Experiments
confirm that our method matches the convergence rate of the parameter error.
