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

TL;DR

This paper introduces a new piecewise linear finite element method for solving an optimal control problem involving fractional elliptic operators on curved domains, with comprehensive error analysis and numerical validation.

Contribution

It develops a novel discretization scheme using piecewise linear functions and anisotropic meshes for fractional operators, extending error analysis to curved domains in any dimension.

Findings

Error estimates match numerical experiments

Method effectively handles curved domains

Applicable in any spatial dimension

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 oder 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 Holder and Sobolev regularity estimates for the optimal variables and error estimates for an 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.