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A priori error analysis for optimal distributed control problem governed by the first order linear hyperbolic equation: hp-streamline diffusion discontinuous galerkin method

  • Chunguang Xiong EMAIL logo , Fusheng Luo , Xiuling Ma and Yu’an Li


In the current paper, we derive the a priori error analysis (hp version) of the streamline diffusion DG finite element approximation for optimal distributed control problem governed by the first order linear hyperbolic equation. We present the stability of such method, obtain the a priori error upper bound for the state and the control approximation, and prove the convergence of numerical method. For the optimal control problem, these estimates are apparently not available in the literature.


The work of the second author was supported in part by the NSFC Grant No. 11401129.


[1] C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport, Ann. Sci. École Norm. Sup., 3(1970), No. 2, 185–233.10.24033/asens.1190Search in Google Scholar

[2] C. Johnson, U. Nävert, and J. Pitkäranta, Finite element methods for linear hyperbolic problems, Comp. Meth. Appl. Mech. Engrg., 45 (1984), 285–312.10.1016/0045-7825(84)90158-0Search in Google Scholar

[3] D. A. French and J. T. King, Approximation of an elliptic control problem by the finite element method, Numer. Funct. Anal. Optim., 12 (1991), 299–314.10.1080/01630569108816430Search in Google Scholar

[4] F. S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), 28–47.10.1016/0022-247X(73)90022-XSearch in Google Scholar

[5] G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), 414–427.10.1137/0320032Search in Google Scholar

[6] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer,1971.10.1007/978-3-642-65024-6Search in Google Scholar

[7] K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control constrained, optimal control systems, Appl. Math. Optim., 8 (1982), 69–95.10.1007/BF01447752Search in Google Scholar

[8] L. Hou and J. C. Turner, Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls, Numer. Math., 71 (1995), 289–315.10.1007/s002110050146Search in Google Scholar

[9] M. Stynes and L. Tobiska, Analysis of the streamline diffusion finite element method for a convection–diffusion problem with exponential layers, East-West J. Numer. Math., 9 (2001), 59–76.Search in Google Scholar

[10] P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, Marcel Dekker, New York, 1994.Search in Google Scholar

[11] P. Houston, C. Schwab, and E.Suli, Discontinuous hp finite element methods for advection–diffusion–reaction problems, SIAM J. Numer. Anal., 39 (2002), No. 6, 2133–2163.10.1137/S0036142900374111Search in Google Scholar

[12] P. Houston, C. Schwab and E. Suli, Stabilized hp finite element method for first-order hyperbolic problems, SIAM. J. Numer. Anal. 37 (2000), No. 5, 1618–1643.10.1137/S0036142998348777Search in Google Scholar

[13] P. Castillo, B. Cockburn, D. Schötzau, and C. Schwab, Optimal a priori error estimates for the hp version of the local discontinuous Galerkin method for convection diffusion problem, Math. Comp. 71 (2001), No. 238, 455–478.10.1090/S0025-5718-01-01317-5Search in Google Scholar

[14] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 6: Evolution Problems II, Springer-Verlag, Berlin, 1992.Search in Google Scholar

[15] R. Li, W. Liu, H. Ma, and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problem, SIAM J. Control Optim., 41 (2002), No. 5, 1321–1349.10.1137/S0363012901389342Search in Google Scholar

[16] T. Geveci, On the approximation of the solution of an optimal control problem governed by an ellpitic equation, RAIRO Anal. Numer. 13 (1979), 313–328.10.1051/m2an/1979130403131Search in Google Scholar

[17] T. Linß and M. Stynes, Numerical methods on Shishkin meshes for linear convection–diffusion problems, Comput. Methods Appl. Mech. Engrg., 190 (2001), 3527–3542.10.1016/S0045-7825(00)00271-1Search in Google Scholar

[18] W. Alt and U. Mackenroth, Convergence of finite element approximation to state constrained convex parabolic boundary control problems, SIAM. J. Control Optim., 27 (1989), 718–736.10.1137/0327038Search in Google Scholar

[19] W. Liu and N. Yan, A posteriori error esimator for convex boundary control problems, SIAM J. Numer. Anal., 39 (2001), 73–99.10.1137/S0036142999352187Search in Google Scholar

[20] F. Tröltzsch, Semidiscrete Ritz–Galerkin approximation of nonlinear parabolic boundary control problems — Strong convergence of optimal controls, Appl. Math. Optim., 29 (1994), No. 3, 309–329.10.1007/BF01189480Search in Google Scholar

[21] C. Xiong, Y. Li, Error analysis for optimal control problem governed by convection–diffusion equations: DG method. J. Comput. Appl. Math., 235 (2011), No. 10, 3163–3177.10.1016/ in Google Scholar

Received: 2014-7-14
Revised: 2015-2-6
Accepted: 2015-3-5
Published Online: 2016-6-8
Published in Print: 2016-6-1

© 2016 by Walter de Gruyter Berlin/Boston

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