Superconvergence for General Convex Optimal Control Problems Governed by Semilinear Parabolic Equations

Abstract

We will investigate the superconvergence for the semidiscrete finite element approximation of distributed convex optimal control problems governed by semilinear parabolic equations. The state and costate are approximated by the piecewise linear functions and the control is approximated by piecewise constant functions. We present the superconvergence analysis for both the control variable and the state variables.

Superconvergence of semidiscrete finite element methods for bilinear parabolic optimal control problems

“…In recent years, there have been considerable related results for finite element approximation of linear or semilinear parabolic optimal control problems (see, e.g., [9–11]). Although bilinear optimal control problems are frequently met in applications, they are much more difficult to handle in comparison to linear or semilinear cases.…”

“…Although there has been extensive research on a priori error estimates and superconvergence of finite element methods for various optimal control problems, it mostly focused on linear or semilinear elliptic cases (see, e.g., [ 2 – 4 , 6 ]). In recent years, there have been considerable related results for finite element approximation of linear or semilinear parabolic optimal control problems (see, e.g., [ 9 – 11 ]). Although bilinear optimal control problems are frequently met in applications, they are much more difficult to handle in comparison to linear or semilinear cases.…”

“…A priori error estimates of space-time finite element and standard finite element approximation for linear parabolic control problem were derived in [ 8 ] and [ 9 ]. The superconvergence of variational discretization and standard finite element approximation for semilinear parabolic control problem can be found in [ 10 ] and [ 11 ], respectively.…”

Convergence and superconvergence of variational discretization for parabolic bilinear optimization problems

“…The convergent order will be improved to O(h 3 2 ) or O(h 3 2 + k) by superconvergence analysis. Some superconvergence results of FEMs for linear and semilinear elliptic or parabolic OCPs can be found in [4,6,15,[27][28][29]. Adaptive FEMs that approximate elliptic and parabolic OCPs have been investigated in [1,11,19,32] and [3], respectively.…”

High accuracy analysis of finite element methods for optimal control problems and its application

“…文献[66] 考虑了抛物最优控制问题的半离散有限元逼近, 得到了 真解的插值与有限元解之间 O(h 2 ) 的超收敛性. 关于具有逐点控制约束和积分控制约束的线性和半 线性抛物方程及双曲方程最优控制问题相关的研究可参见文献[67][68][69][70].…”