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

Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile .
enrique.otarola@usm.cl

Received: 11 August 2015
Revised: 6 October 2016
Accepted: 10 October 2016

Abstract

We propose and analyze a new discretization technique for a linear-quadratic optimal control problem involving the fractional powers of a symmetric and uniformly elliptic second order operator; control constraints are considered. Since these fractional operators can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic equation, we recast our problem as a nonuniformly elliptic optimal control problem. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We propose a fully discrete scheme that is based on piecewise linear functions on quasi-uniform meshes to approximate the optimal control and first-degree tensor product functions on anisotropic meshes for the optimal state variable. We provide an a priori error analysis that relies on derived Hölder and Sobolev regularity estimates for the optimal variables and error estimates for a scheme that approximates fractional diffusion on curved domains; the latter being an extension of previous available results. The analysis is valid in any dimension. We conclude by presenting some numerical experiments that validate the derived error estimates.