An adaptive finite element method for distributed elliptic optimal control problems with variable energy regularization

TL;DR

This paper develops an adaptive finite element method for elliptic optimal control problems with variable energy regularization, achieving optimal error estimates and demonstrating efficiency and robustness through numerical experiments.

Contribution

It introduces a novel adaptive scheme using a mesh-dependent regularization parameter and analyzes its optimal error behavior for discontinuous targets.

Findings

Error between computed and target states is optimal with mesh-dependent regularization.

Adaptive scheme effectively handles discontinuous target functions.

Numerical results confirm theoretical error estimates and solver robustness.

Abstract

We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual $L^2(\Omega)$ norm regularization term with a constant regularization parameter $\varrho$ is replaced by a suitable representation of the energy norm in $H^{-1}(\Omega)$ involving a variable, mesh-dependent regularization parameter $\varrho(x)$. It turns out that the error between the computed finite element state $\widetilde{u}_{\varrho h}$ and the desired state $\overline{u}$ (target) is optimal in the $L^2(\Omega)$ norm provided that $\varrho(x)$ behaves like the local mesh size squared. This is especially important when adaptive meshes are used in order to approximate discontinuous target functions. The adaptive scheme can be driven by the computable and localizable error norm $\| \widetilde{u}_{\varrho h} - \overline{u}\|_{L^2(\Omega)}$ between the finite element state $\widetilde{u}_{\varrho h}$ and the target $\overline{u}$. The numerical results not only illustrate our theoretical findings, but also show that the iterative solvers for the discretized reduced optimality system are very efficient and robust.