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

Otárola, Enrique

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1473-1500.

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.

Classification : 35R11, 35J70, 49J20, 49M25, 65N12, 65N30
Mots clés : Linear-quadratic optimal control problem, fractional diffusion, finite elements, anisotropic estimates, curved domains

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