Adaptive Finite Element Approximation of Sparse Optimal Control Problem with Integral Fractional Laplacian

In this paper, we present and analyze a weighted residual a posteriori error estimate for a sparse optimal control problem. The problem involves a non-differentiable cost functional, a state equation with an integral fractional Laplacian, and control constraints. We employ subdifferentiation in non-differentiable 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 by the variational discretization method. Upper bounds for the a posteriori error estimate of the finite element approximation of the optimal control problem are derived. One challenge in devising a posteriori error estimators is poor properties of the residual. Namely, it is not necessarily in \(L^2(\varOmega )\) $L^2(\Omega )$ . To address this issue, the weighted residual estimator incorporates additional weight computed as the power of the distance from the mesh skeleton. Furthermore, we propose an h-adaptive algorithm driven by the a posteriori error estimator, utilizing the Dörfler 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.