Adaptive FEM for parameter-errors in elliptic linear-quadratic parameter estimation problems
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2021
- DOI:
- arXiv:
- arXiv:2111.03627
- Bibcode:
- 2021arXiv211103627B
- Keywords:
-
- Mathematics - Numerical Analysis;
- Mathematics - Optimization and Control
- E-Print:
- doi:10.1137/21M1458077