A FEM for an optimal control problem of fractional powers of elliptic operators

TL;DR

This paper develops and analyzes finite element methods for solving a linear-quadratic optimal control problem involving fractional elliptic operators, using a truncated domain approach and anisotropic meshes for improved numerical accuracy.

Contribution

It introduces a truncated domain formulation for fractional elliptic operators and compares semi-discrete and fully discrete finite element approaches with error analysis.

Findings

Anisotropic meshes outperform quasi-uniform meshes in accuracy.

Derived a priori error estimates for both control discretization methods.

Numerical results confirm the effectiveness of the proposed methods.

Abstract

We study solution techniques for a linear-quadratic optimal control problem involving fractional powers of elliptic operators. These fractional operators can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. Thus, we consider an equivalent formulation with a nonuniformly elliptic operator as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We discretize the proposed truncated state equation using first degree tensor product finite elements on anisotropic meshes. For the control problem we analyze two approaches: one that is semi-discrete based on the so-called variational approach, where the control is not discretized, and the other one is fully discrete via the discretization of the control by piecewise constant functions. For both approaches, we derive a priori error estimates with respect to the degrees of freedom. Numerical experiments validate the derived error estimates and reveal a competitive performance of anisotropic over quasi-uniform refinement.