In this paper, we establish error estimates for the numerical approximation of the parabolic optimal control problem with measure data in a two-dimensional nonconvex polygonal domain. Due to the presence of measure data in the state equation and the nonconvex nature of the domain, the finite element error analysis is not straightforward. Regularity results for the control problem based on the first-order optimality system are discussed. The state variable and co-state variable are approximated by continuous piecewise linear finite element, and the control variable is approximated by piecewise constant functions. A priori error estimates for the state and control variable are derived for spatially discrete control problem and fully discrete control problem in L 2 (L 2 )-norm. A numerical experiment is performed to illustrate our theoretical findings.
We analyze both a priori and a posteriori error analysis of finite-element method for elliptic optimal control problems with measure data in a bounded convex domain in R d (d = 2 or 3). The solution of the state equation of such type of problems exhibits low regularity due to the presence of measure data, which introduces some difficulties for both theory and numerics of the finite-element method. We first prove the existence, uniqueness, and regularity of the solution to the optimal control problem. To discretize the control problem, we use continuous piecewise linear elements for the approximations of the state and co-state variables, whereas piecewise constant functions are used for the control variable.We derive a priori error estimates of order (h 2− d 2 ) for the state, co-state, and control variables in the L 2 -norm. Further, global a posteriori upper bounds for the state, co-state, and control variables in the L 2 -norm are established. Moreover, local lower bounds for the errors in the state and co-state variables and a global lower bound for the error in the control variable are obtained in the case of two space dimensions (d = 2). Numerical experiments are provided, which support our theoretical results. KEYWORDS a priori and a posteriori error estimates, elliptic optimal control problem, finite-element approximations, measure data Optim Control Appl Meth. 2019;40:241-264.wileyonlinelibrary.com/journal/oca
Summary
In this exposition, we study both a priori and a posteriori error analysis for the H1‐Galerkin mixed finite element method for optimal control problems governed by linear parabolic equations. The state and costate variables are approximated by the lowest order Raviart‐Thomas finite element spaces, whereas the control variable is approximated by piecewise constant functions. Compared to the standard mixed finite element procedure, the present method is not subject to the Ladyzhenskaya‐Babuska‐Brezzi condition and the approximating finite element spaces are allowed to be of different degree polynomials. A priori error analysis for both the semidiscrete and the backward Euler fully discrete schemes are analyzed, and
L∞false(L2false) convergence properties for the state variables and the control variable are obtained. In addition, L2(L2)‐norm a posteriori error estimates for the state and control variables and
L∞false(L2false)‐norm for the flux variable are also derived.
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