A stabilized finite element method for the convection dominated diffusion optimal control problem

References

• Hughes TJR, Francea LP, Hulbert GM. A new finite element formulation for computational fluid dynamics: Viii. The Galerkin-least-square method for advective-diffusive equations. Comput. Methods Appl. Mech. Eng. 1989;73:173–189.
   Web of Science ®Google Scholar
• Baiocchi C, Brezzi F, Francea LP. Virtual bubbles and Galerkin-least-square type methods. Comput. Methods Appl. Mech. Eng. 1993;105:125–141.
   Web of Science ®Google Scholar
• Brooks AN, Hughes TJR. Streamline upwind Petrov--Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier--Stokes equations. Comput. Methods Appl. Mech. Eng. 1982;32:199–259.
   Web of Science ®Google Scholar
• Russo A. Stream-upwind Petrov/Galerkin method (SUPG) vs residual-free bubbles (RFB). Comput. Methods Appl. Mech. Eng. 2006;195:1608–1626.
   Web of Science ®Google Scholar
• Brezzi F, Russo A. Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 1994;4:571–587.
   Web of Science ®Google Scholar
• Franca LP, Russo A. Deriving upwinding mass lumping and selective reduced integration by residual-free bubbles. Appl. Math. Lett. 1996;9:83–88.
   Web of Science ®Google Scholar
• Braack M, Burman E. Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 2006;43:2544–2566.
   Web of Science ®Google Scholar
• Matthies G, Skrzypacz P, Tobiska L. A unified convergence analysis for local projection stabilisations applied to the Oseen problem. M2AN Math. Model. Numer. Anal. 2007;41:713–742.
   Web of Science ®Google Scholar
• Zheng HB, Hou YR, Shi F. Adaptive variational multiscale methods for incompressible flows based on two local Gauss integrations. J. Comput. Phys. 2010;229:7030–7041.
   Web of Science ®Google Scholar
• Burman E, Hansbo P. Edge stabilization for Galerkin approximations of convection-diffusion reaction problems. Comput. Method. Appl. Mech. Eng. 2004;193:1437–1453.
   Web of Science ®Google Scholar
• Knobloch P. A generalization of the local projection stabilization for convection--diffusion--reaction equations. SIAM J. Numer. Anal. 2010;48:659–680.
   Web of Science ®Google Scholar
• Wu X, Dai Y, Tao W. MLPG/SUPG method for convection-dominated problems. Numer. Heat Transfer B. 2012;61:36–51.
   Web of Science ®Google Scholar
• Song LN, Hou YR, Zheng HB. A variational multiscale method based on bubble functions for convection-dominated convection-diffusion equation. Appl. Math. Comput. 2010;217:2226–2237.
   Web of Science ®Google Scholar
• Becker R, Vexler B. Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 2007;106:349–367.
   Web of Science ®Google Scholar
• Yan N, Zhou Z. A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection dominated diffusion equation. J. Comput. Appl. Math. 2009;223:198–217.
   Web of Science ®Google Scholar
• Zhou Z, Yan N. The local discontinuous Galerkin method for optimal control problem governed by convection diffusion equations. Int. J. Numer. Anal. Model. 2010;7:681–699.
   Web of Science ®Google Scholar
• Y{\"u}cel H, Heinkenschloss M, Karasözen B. Distributed optimal control of diffusion-convection reaction equations using discontinuous Galerkin methods. In: Numerical mathematics and advanced applications. 2011. Berlin: Springer; 2013. p. 389–397.
   Google Scholar
• Akman T, Yücel H, Karasözen B. A priori error analysis of the upwind symmetric interior penalty Galerkin (SIPG) method for the optimal control problems governed by unsteady convection diffusion equations. Comput. Optim. Appl. 2014;57:703–729.
   Web of Science ®Google Scholar
• Hinze M. A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 2005;30:45–61.
   Web of Science ®Google Scholar
• Hinze M, Yan N, Zhou Z. Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math. 2009;27:237–253.
   Web of Science ®Google Scholar
• Tr{\"o}ltzsch F. Optimal control of partial differential equations: theory, methods, and applications. In: Graduate studies in mathematics. Vol. 112. American Mathematical Society; 2010.
   Google Scholar
• Kunisch K, Lu X. Optimal control for elliptic systems with pointwise euclidean norm constraints on the controls. Math. Program. 2013;142:461–483.
   Web of Science ®Google Scholar
• Weng Z, Yang JZ, Lu X. Two-level quadratic equal-order stabilized method for the Stokes eigenvalue problem. Int. J. Comput. Math. 2015;92:337–448.
   Web of Science ®Google Scholar
• Ciarlet PG. The finite element method for elliptic problems. Amsterdam: North-Holland; 1978.
   Google Scholar
• Hintermüller M, Ito K, Kunisch K. The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 2003;13:865–888.
   Web of Science ®Google Scholar
• John V, Kaya S, Layton W. A two-level variational multiscale method for convection-dominated convection-diffusion equations. Comput. Methods Appl. Mech. Eng. 2006;195:4594–4603.
   Web of Science ®Google Scholar
• Ito K, Kunisch K. Lagrange multiplier approach to variational problems and applications.. Vol. 15, Advances in Design and Control. Philadelphia (PA): SIAM; 2008.
   Google Scholar
• Kunisch K, Lu X. Optimal control for an elliptic system with convex polygonal control constraints. IMA J. Numer Anal. 2013;33:875–897.
   Web of Science ®Google Scholar