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

TL;DR

This paper introduces an adaptive finite element method for elliptic linear-quadratic parameter estimation problems, providing a new a priori bound and an estimator that captures the faster convergence of parameter errors, with proven optimal rates.

Contribution

It presents a novel a priori bound and an adaptive FEM driven by an a posteriori estimator that accurately reflects the convergence behavior of parameter errors in elliptic problems.

Findings

The estimator decreases at a rate equal to the sum of the best approximation rates of state and co-state variables.

The method achieves optimal convergence rates for parameter errors.

Experiments confirm the estimator's effectiveness in matching the convergence rate of the parameter error.

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.