Superconvergence of quadratic optimal control problems by triangular mixed finite element methods

Abstract

SUMMARYThe aim of this work is to investigate the discretization of a quadratic convex optimal control problem using the mixed finite element method. The state and co-state are approximated by the order k 1 RaviartThomas mixed finite element spaces, and the control is approximated by piecewise constant functions. We construct an interpolation of the exact control and a projection of the discrete scalar co-state to be the approximated solution of the control variable for the continuous optimal control problem. As a result, it can be proved that the difference between the interpolation and the piecewise constant approximation has superconvergence property for the control of order h 3/2 for k = 0 and of order h 2 for k = 1. Moreover, only for the order k = 1 Raviart-Thomas mixed finite element approximation does the postprocessing technique possess the superconvergence property of order h 2 . Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

A priori error estimates of mixed finite element methods for general semilinear elliptic optimal control problems

“…Recently, we have made some primary studies on a priori error estimates and superconvergence for linear optimal control problems by mixed finite element methods in [5,6,11,10,23]. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation.…”

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

“…Recently, we obtained a priori error estimates and a posteriori error estimates of mixed finite element methods for linear and nonlinear optimal control problems [7][8][9] . 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] .…”

“…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…”

A POSTERIORI L∞(L2)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL 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 [16,17,18] the authors have done some primary works on a priori, superconvergence and a posteriori error estimates error estimates for linear elliptic optimal control problems by mixed finite element methods. However, there doesn't seem to exist any work on a posteriori error analysis of mixed finite element approximation for hyperbolic problems in the literature, especially for hyperbolic optimal control problems.…”