Superconvergence of mixed finite element methods for optimal control problems

Abstract

Abstract. In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique. We derive L 2 superconvergence properties for the flux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections. Moreover, global L 2 superconvergence results are obtained by virtue of an interpolation postprocessing technique. Thus, based on these superconvergence estimates, some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

Superconvergence of triangular mixed finite element methods for nonlinear optimal control problems

“…Then we used the postprocessing projection operator to prove a quadratic superconvergence of the control for linear elliptic optimal control problem by a mixed finite element method [10][11][12] . We are concerned with the 2-d nonlinear elliptic optimal control problem…”

Error Estimates of Rt1 Mixed Methods for Distributed Optimal Control Problems

“…For the mixed finite element approximations of optimal control problems, Chen et al have done some works on priori error estimates and superconvergence properties of mixed finite elements for optimal control problems, see [4,5,7,8]. Recently, in [22], Xing and Chen have analyzed the L ∞ -error estimates for general convex optimal control problems with the lowest order Raviart-Thomas mixed finite element methods, while the L ∞ -error estimates for quadratic optimal control problems governed by semilinear elliptic equations was investigated in [18].…”

Error Estimates of Mixed Finite Element Approximations for a Class of Fourth Order Elliptic Control Problems

“…When the objective functional contains the gradient of the state variable, mixed finite element methods should be used for discretization of the state equation with which both the scalar variable and its flux variable can be approximated in the same accuracy. Recently, in [5,6,7,8], Chen et al have done some primary works on a priori, superconvergence and a posteriori error estimates for linear elliptic optimal control problems by mixed finite element methods.…”