SUPERCONVERGENCE ANALYSIS OF FINITE ELEMENT METHODS FOR OPTIMAL CONTROL PROBLEMS OF THE STATIONARY B

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Benard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in L 1 -norm and optimal error estimates in L 2 -norm.

[1]  Danping Yang,et al.  TWO FINITE-ELEMENT SCHEMES FOR STEADY CONVECTIVE HEAT TRANSFER WITH SYSTEM ROTATION AND VARIABLE THERMAL PROPERTIES , 2005 .

[2]  M. Křížek,et al.  INTERNAL FE APPROXIMATION OF SPACES OF DIVERGENCE-FREE FUNCTIONS IN THREE-DIMENSIONAL DOMAINS , 1986 .

[3]  T. Geveci,et al.  On the approximation of the solution of an optimal control problem governed by an elliptic equation , 1979 .

[4]  Dan Tiba,et al.  ERROR ESTIMATES IN THE APPROXIMATION OF OPTIMIZATION PROBLEMS GOVERNED BY NONLINEAR OPERATORS , 2001 .

[5]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[6]  Fredi Tröltzsch,et al.  Error estimates for linear-quadratic elliptic control problems , 2002, Analysis and Optimization of Differential Systems.

[7]  L. Steven Hou,et al.  Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls , 1995 .

[8]  Michael Hinze,et al.  A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case , 2005, Comput. Optim. Appl..

[9]  Frédéric Abergel,et al.  Some optimal control problems of multistate equations appearing in fluid mechanics , 1993 .

[10]  L. Hou,et al.  Boundary velocity control of incompressible flow with an application to viscous drag reduction , 1992 .

[11]  Ruo Li,et al.  A Posteriori Error Estimates of Recovery Type for Distributed Convex Optimal Control Problems , 2007, J. Sci. Comput..

[12]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[13]  Ningning Yan,et al.  GLOBAL SUPERCONVERGENCE FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY STOKES EQUATIONS , 2006 .

[14]  Wenbin Liu,et al.  Local A Posteriori Error Estimates for Convex Boundary Control Problems , 2009, SIAM J. Numer. Anal..

[15]  Wenbin Liu,et al.  A Posteriori Error Estimates for Convex Boundary Control Problems , 2001, SIAM J. Numer. Anal..

[16]  Fredi Tröltzsch,et al.  Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem , 2002, Comput. Optim. Appl..

[17]  Fredi Tröltzsch,et al.  NECESSARY AND SUFFICIENT OPTIMALITY CONDITIONS FOR OPTIMIZATION PROBLEMS IN FUNCTION SPACES AND APPLICATIONS TO CONTROL THEORY , 2003 .

[18]  L. Hou,et al.  Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls , 1991 .

[19]  L. A. Rukhovets,et al.  Study of the rate of convergence of variational difference schemes for second-order elliptic equations in a two-dimensional field with a smooth boundary , 1969 .

[20]  Marion Kee,et al.  Analysis , 2004, Machine Translation.

[21]  Max Gunzburger,et al.  Analysis, approximation, and computation of a coupled solid/fluid temperature control problem , 1994 .

[22]  K. Malanowski.  Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems , 1982 .

[23]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[24]  Richard S. Falk,et al.  Approximation of a class of optimal control problems with order of convergence estimates , 1973 .

[25]  Hyung-C. Lee,et al.  Analysis of Optimal Control Problems for the 2-D Stationary Boussinesq Equations , 2000 .

[26]  Arnd Rösch,et al.  Superconvergence Properties of Optimal Control Problems , 2004, SIAM J. Control. Optim..

[27]  C. Cutelier,et al.  Optimal Control of a System Govebned by the Navier-Stokes Equations Coupled with the Heat Equation , 1976 .

[28]  G. V. Alekseev,et al.  Solvability of stationary boundary control problems for heat convection equations , 1998 .

[29]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[30]  Fredi Tröltzsch,et al.  Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems , 2005, Comput. Optim. Appl..

[31]  Jacques Rappaz,et al.  Finite Dimensional Approximation of Non-Linear Problems .1. Branches of Nonsingular Solutions , 1980 .