Local discontinuous galerkin approximation of convection‐dominated diffusion optimal control problems with control constraints

In this article, we investigate local discontinuous Galerkin approximation of stationary convection‐dominated diffusion optimal control problems with distributed control constraints. The state variable and adjoint state variable are approximated by piecewise linear polynomials without continuity requirement, whereas the control variable is discretized by variational discretization concept. The discrete first‐order optimality condition is derived. We show that optimization and discretization are commutative for the local discontinuous Galerkin approximation. Because the solutions to convection‐dominated diffusion equations often admit interior or boundary layers, residual type a posteriori error estimate in L2 norm is proved, which can be used to guide mesh refinement. Finally, numerical examples are presented to illustrate the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 339–360, 2014

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

[2]  Ivo Babuška,et al.  The optimal convergence rate of the p-version of the finite element method , 1987 .

[3]  M. Stynes,et al.  Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems , 1996 .

[4]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[5]  Aurea Martínez,et al.  Theoretical and Numerical Analysis of an Optimal Control Problem Related to Wastewater Treatment , 2000, SIAM J. Control. Optim..

[6]  Ilaria Perugia,et al.  An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems , 2000, SIAM J. Numer. Anal..

[7]  S. Scott Collis,et al.  Analysis of the Streamline Upwind/Petrov Galerkin Method Applied to the Solution of Optimal Control Problems ∗ , 2002 .

[8]  Jiang Zhu,et al.  A mathematical formulation for optimal control of air pollution , 2003 .

[9]  Martin Stynes,et al.  Steady-state convection-diffusion problems , 2005, Acta Numerica.

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

[11]  Alfio Quarteroni,et al.  Optimal Control and Numerical Adaptivity for Advection-Diffusion Equations , 2005 .

[12]  Roland Becker,et al.  Optimal control of the convection-diffusion equation using stabilized finite element methods , 2007, Numerische Mathematik.

[13]  Zhaojie,et al.  A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation , 2008 .

[14]  Roland Glowinski,et al.  A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics , 2008 .

[15]  Ningning Yan,et al.  A RT Mixed FEM/DG Scheme for Optimal Control Governed by Convection Diffusion Equations , 2009, J. Sci. Comput..

[16]  Wenbin Liu,et al.  DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD WITH INTERIOR PENALTIES FOR CONVECTION DIFFUSION OPTIMAL CONTROL PROBLEM , 2009 .

[17]  Zhaojie,et al.  VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS , 2009 .

[18]  Ningning Yan,et al.  A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation , 2009 .

[19]  Zhaojie Zhou,et al.  THE LOCAL DISCONTINUOUS GALERKIN METHOD FOR OPTIMAL CONTROL PROBLEM GOVERNED BY CONVECTION DIFFUSION EQUATIONS , 2010 .

[20]  Matthias Heinkenschloss,et al.  Local Error Estimates for SUPG Solutions of Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems , 2010, SIAM J. Numer. Anal..

[21]  Hans-Görg Roos,et al.  Numerical Analysis of a System of Singularly Perturbed Convection-Diffusion Equations Related to Optimal Control , 2011 .

[22]  Yuan Li,et al.  Error analysis for optimal control problem governed by convection diffusion equations: DG method , 2011, J. Comput. Appl. Math..

[23]  Hans-Görg Roos,et al.  Robust Numerical Methods for Singularly Perturbed Differential Equations: A Survey Covering 2008–2012 , 2012 .

[24]  Dmitriy Leykekhman,et al.  Investigation of Commutative Properties of Discontinuous Galerkin Methods in PDE Constrained Optimal Control Problems , 2012, J. Sci. Comput..

[25]  R. Hoppe,et al.  Weak-duality based adaptive finite element methods for PDE-constrained optimization with pointwise gradient state-constraints , 2012 .

[26]  Matthias Heinkenschloss,et al.  Local Error Analysis of Discontinuous Galerkin Methods for Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems , 2012, SIAM J. Numer. Anal..

[27]  Bülent Karasözen,et al.  Distributed Optimal Control of Diffusion-Convection-Reaction Equations Using Discontinuous Galerkin Methods , 2013 .

[28]  Hongxing Rui,et al.  Adaptive characteristic finite element approximation of convection–diffusion optimal control problems , 2013 .