Superconvergence of fully discrete rectangular mixed finite element methods of parabolic control problems

Convergence and superconvergence of variational discretization for parabolic bilinear optimization problems

“…Although there has been extensive research on convergence and superconvergence of FEMs for various parabolic OCPs, mostly focused on linear or semilinear parabolic cases (see, e.g., [6,10,16,26,30]), the results on convergence and superconvergence are O(h + k) and O(h 3 2 + k), respectively. Recent years, VD are used to deal with different OCPs in [7,13,14].…”

“…It is well known that optimal control and optimization problems are approximated by many numerical methods, such as standard finite element methods (FEMs), mixed FEMs, space-time FEMs, finite volume element methods, spectral methods, multigrid methods etc. ; see e.g., [5,8,10,16,17,[24][25][26]31]. There is no doubt that FEMs occupy the most important position in these methods.…”

Superconvergence of $H$¹-Galerkin Mixed Finite Element Methods for Elliptic Optimal Control Problems

“…Recently, Chen et al [3,4,7,15] studied a priori error estimates and superconvergence of the Raviart-Thomas mixed finite element method for elliptic and parabolic optimal control problems. In particular, to show the superconvergence of the control, the postprocessing projection operator, introduced by Meyer and Rösch [22], has been used in [3,4] and the average L 2 projection operator in [7].…”

“…However, the low regularity of the control implies the convergence order h 3/2 . Hou and Chen [15] discussed the superconvergence of fully discrete mixed finite element methods for parabolic optimal control problems and presented two results for the control variable derived by the use of a recovery operator and a postprocessing projection operator.…”