A priori and a posteriori error estimates of finite‐element approximations for elliptic optimal control problem with measure data. (19th November 2018)

• Record Type: Journal Article
• Authors: Shakya, Pratibha; Sinha, Rajen Kumar
• Is Part Of: Optimal control applications and methods. Volume 40:Number 2(2019)
• Page Start: 241
• Page End: 264
• Publication Date: 2018-11-19
• Subjects: a priori and a posteriori error estimates -- elliptic optimal control problem -- finite‐element approximations -- measure data; Control theory -- Periodicals; Mathematical optimization -- Periodicals; 629.8312
• DOI: 10.1002/oca.2476
• Languages: English
• ISSNs: 0143-2087

Abstract

Summary: 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 = 2or3). 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 O ( 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.