An a posteriori error analysis for an optimal control problem involving the fractional Laplacian

TL;DR

This paper develops an efficient a posteriori error estimator for a fractional Laplacian optimal control problem, enabling adaptive methods that achieve optimal convergence without restrictive domain assumptions.

Contribution

It introduces a reliable a posteriori error estimator for fractional Laplacian control problems, relaxing previous domain and data compatibility restrictions.

Findings

The error estimator effectively guides adaptive refinement.

The adaptive strategy achieves optimal convergence rates.

The approach is valid in any spatial dimension.

Abstract

In a previous work, we introduced a discretization scheme for a constrained optimal control problem involving the fractional Laplacian. For such a control problem, we derived optimal a priori error estimates that demand the convexity of the domain and some compatibility conditions on the data. To relax such restrictions, in this paper, we introduce and analyze an efficient and, under certain assumptions, reliable a posteriori error estimator. We realize the fractional Laplacian as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi--infinite cylinder in one more spatial dimension. This extra dimension further motivates the design of an posteriori error indicator. The latter is defined as the sum of three contributions, which come from the discretization of the state and adjoint equations and the control variable. The indicator for the state and adjoint equations relies on an anisotropic error estimator in Muckenhoupt weighted Sobolev spaces. The analysis is valid in any dimension. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that exhibits optimal experimental rates of convergence.