Error analysis for parabolic optimal control problems with measure data in a nonconvex polygonal domain

TL;DR

This paper analyzes finite element approximations for parabolic optimal control problems with measure data in nonconvex polygonal domains, addressing low regularity and providing error bounds validated by numerical experiments.

Contribution

It establishes existence, uniqueness, and regularity results, and derives error estimates for finite element methods applied to these challenging control problems.

Findings

Finite element approximations converge at lower orders due to low regularity.

A priori and a posteriori error bounds are derived for state, control, and co-state variables.

Numerical experiments confirm the theoretical convergence rates.

Abstract

This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain. Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain. The low regularity of the solution allows the finite element approximations to converge at lower orders. We prove the existence, uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition. For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables, whereas piecewise constant functions are employed to approximate the control variable. The temporal discretization is based on the implicit Euler scheme. We derive both a priori and a posteriori error bounds for the state, control and co-state variables. Numerical experiments are performed to validate the theoretical rates of convergence.