Error estimates for fractional semilinear optimal control on Lipschitz polytopes

TL;DR

This paper investigates error estimates for fractional semilinear elliptic optimal control problems on Lipschitz polytopes, comparing semidiscrete and fully discrete finite element schemes with new theoretical bounds.

Contribution

It introduces improved error bounds for finite element discretizations of fractional semilinear elliptic PDEs on Lipschitz polytopes, enhancing existing theoretical results.

Findings

Derived error estimates for fractional semilinear elliptic PDEs on graded meshes

Established error bounds for semidiscrete control schemes

Improved existing error bounds for fully discrete schemes

Abstract

We adopt the integral definition of the fractional Laplace operator and analyze solution techniques for fractional, semilinear, and elliptic optimal control problems posed on Lipschitz polytopes. We consider two strategies of discretization: a semidiscrete scheme where the admissible control set is not discretized and a fully discrete scheme where such a set is discretized with piecewise constant functions. As an instrumental step, we derive error estimates for finite element discretizations of fractional semilinear elliptic partial differential equations (PDEs) on quasi-uniform and graded meshes. With these estimates at hand, we derive error bounds for the semidiscrete scheme and improve the ones that are available in the literature for the fully discrete scheme.