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

Adaptive FEM for parameter-errors in elliptic linear-quadratic parameter estimation problems