A piecewise linear FEM for an optimal control problem of fractional operators: error analysis on curved domains

摘要

We propose and analyze a new discretization technique for a linear-quadraticoptimal control problem involving the fractional powers of a symmetric anduniformly elliptic second oder operator; control constraints are considered.Since these fractional operators can be realized as the Dirichlet-to-Neumannmap for a nonuniformly elliptic equation, we recast our problem as anonuniformly elliptic optimal control problem. The rapid decay of the solutionto this problem suggests a truncation that is suitable for numericalapproximation. We propose a fully discrete scheme that is based on piecewiselinear functions on quasi-uniform meshes to approximate the optimal control andfirst-degree tensor product functions on anisotropic meshes for the optimalstate variable. We provide an a priori error analysis that relies on derivedHolder and Sobolev regularity estimates for the optimal variables and errorestimates for an scheme that approximates fractional diffusion on curveddomains; the latter being an extension of previous available results. Theanalysis is valid in any dimension. We conclude by presenting some numericalexperiments that validate the derived error estimates.