Adaptive finite element approximation of sparse optimal control with integral fractional Laplacian

TL;DR

This paper develops and analyzes an adaptive finite element method with a posteriori error estimates for an optimal control problem involving a fractional Laplacian, ensuring optimal convergence rates.

Contribution

It introduces a weighted residual a posteriori error estimator for fractional Laplacian control problems and proves its effectiveness with an adaptive algorithm.

Findings

The adaptive algorithm converges at the optimal algebraic rate.

The weighted residual estimator effectively bounds the error.

Numerical experiments confirm theoretical convergence rates.

Abstract

In this paper we present and analyze a weighted residual a posteriori error estimate for an optimal control problem. The problem involves a nondifferentiable cost functional, a state equation with an integral fractional Laplacian, and control constraints. We employ subdifferentiation in the context of nondifferentiable convex analysis to obtain first-order optimality conditions. Piecewise linear polynomials are utilized to approximate the solutions of the state and adjoint equations. The control variable is discretized using the variational discretization method. Upper and lower bounds for the a posteriori error estimate of the finite element approximation of the optimal control problem are derived. In the region where 3/2 < alpha < 2, the residuals do not satisfy the L2(Omega) regularity. To address this issue, an additional weight is included in the weighted residual estimator, which is based on a power of the distance from the mesh skeleton. Furthermore, we propose an h-adaptive algorithm driven by the posterior view error estimator, utilizing the Dorfler labeling criterion. The convergence analysis results show that the approximation sequence generated by the adaptive algorithm converges at the optimal algebraic rate. Finally, numerical experiments are conducted to validate the theoretical results.