A first-degree FEM for an optimal control problem of fractional operators: error analysis

Enrique Otarola

Published 2015-08-12Version 1

We study a discretization technique for a linear-quadratic optimal control problem involving fractional diffusion of order $s \in (0,1)$. Since fractional diffusion 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 suggests a truncation that is suitable for numerical approximation. We discretize the truncated problem with a fully discrete scheme based on a piecewise linear finite element approximation on quasi-uniform meshes for the optimal control. The state variable is approximated via first--degree tensor product finite elements on anisotropic meshes. Based on derived H\"older and Sobolev regularity results for the optimal control, we develop an a priori error analysis for $s \in (0,1)$. Numerical experiments validate the derived error estimates.