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R. Becker, M. Innerberger, D. Praetorius:
"Adaptive FEM for parameter-errors in elliptic linear-quadratic parameter estimation problems";
SIAM Journal on Numerical Analysis, 60 (2022), 3; 1450 - 1471.

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.

http://dx.doi.org/10.1137/21M1458077