TY - JOUR
T1 - Error Analysis for Parabolic Optimal Control Problems with Measure Data in a Nonconvex Polygonal Domain
AU - Shakya , Pratibha
JO - Journal of Computational Mathematics
VL - 6
SP - 1579
EP - 1604
PY - 2024
DA - 2024/11
SN - 42
DO - http://doi.org/10.4208/jcm.2305-m2022-0215
UR - https://global-sci.org/intro/article_detail/jcm/23508.html
KW - A priori and a posteriori error estimates, Finite element method, Measure
data, Nonconvex polygonal domain, Optimal control problem.
AB - <p style="text-align: justify;">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.</p>