Year: 2015
Type: article
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 the 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 semidiscrete based on the so-called variational approach, where the control is not discretized, and the other one that 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 degrees of freedom. Numerical experiments validate the derived error estimates and reveal a competitive performance of anisotropic over quasi-uniform refinement.
DOI: 10.1137/140975061
Authors: Harbir Antil, Enrique Otárola
Institution: Universidad Técnica Federico Santa María
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W85517299