2016/03/29 by Harbir Antil, Antil, Harbir, Enrique Otárola +1
Computer Science · Engineering · #35J70 #35R11 #49J20 #49M25 #65N12 #65N30 #65N50 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1603.08989
openalex publication_date 2016/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a previous work, we introduced a discretization scheme for a constrained\noptimal control problem involving the fractional Laplacian. For such a control\nproblem, we derived optimal a priori error estimates that demand the convexity\nof the domain and some compatibility conditions on the data. To relax such\nrestrictions, in this paper, we introduce and analyze an efficient and, under\ncertain assumptions, reliable a posteriori error estimator. We realize the\nfractional Laplacian as the Dirichlet-to-Neumann map for a nonuniformly\nelliptic problem posed on a semi--infinite cylinder in one more spatial\ndimension. This extra dimension further motivates the design of an posteriori\nerror indicator. The latter is defined as the sum of three contributions, which\ncome from the discretization of the state and adjoint equations and the control\nvariable. The indicator for the state and adjoint equations relies on an\nanisotropic error estimator in Muckenhoupt weighted Sobolev spaces. The\nanalysis is valid in any dimension. On the basis of the devised a posteriori\nerror estimator, we design a simple adaptive strategy that exhibits optimal\nexperimental rates of convergence.\n