Volume 51, Number 4, July-August 2017
|Page(s)||1473 - 1500|
|Published online||21 July 2017|
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 .
Received: 11 August 2015
Revised: 6 October 2016
Accepted: 10 October 2016
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.
Mathematics Subject Classification: 35R11 / 35J70 / 49J20 / 49M25 / 65N12 / 65N30
Key words: Linear-quadratic optimal control problem / fractional diffusion / finite elements / anisotropic estimates / curved domains
© EDP Sciences, SMAI 2017
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