Adaptive finite element method for elliptic optimal control problems: convergence and optimality

TL;DR

This paper proves the convergence and optimality of an adaptive finite element method for elliptic optimal control problems with control constraints, using variational discretization and numerical validation.

Contribution

It extends AFEM convergence theory to elliptic optimal control problems with pointwise constraints, demonstrating quasi-optimality and providing rigorous proofs.

Findings

Proves convergence of AFEM for constrained optimal control problems.

Establishes quasi-optimality of the adaptive method.

Numerical experiments confirm theoretical results.

Abstract

In this paper we consider the convergence analysis of adaptive finite element method for elliptic optimal control problems with pointwise control constraints. We use variational discretization concept to discretize the control variable and piecewise linear and continuous finite elements to approximate the state variable. Based on the well-established convergence theory of AFEM for elliptic boundary value problems, we rigorously prove the convergence and quasi-optimality of AFEM for optimal control problems with respect to the state and adjoint state variables, by using the so-called perturbation argument. Numerical experiments confirm our theoretical analysis.