A posteriori error estimates of constrained optimal control problem governed by convection diffusion equations

Abstract

In this paper, we study a posteriori error estimates of the edge stabilization Galerkin method for the constrained optimal control problem governed by convection-dominated diffusion equations. The residual-type a posteriori error estimators yield both upper and lower bounds for control u measured in L²-norm and for state y and costate p measured in energy norm. Two numerical examples are presented to illustrate the effectiveness of the error estimators provided in this paper.

References

1. Ainsworth M, Oden J T . A Posteriori Error Estimationin Finite Element Analysis. New York: Wiley Interscience, 2000
2. Alaoui L El, Ern A, Burman E . A priori and a posteriori analysis of nonconforming finiteelements with face penalty for advection-diffusion equations. IMA J Numer Anal, 2006, 27(1): 151–171. doi:10.1093/imanum/drl011
3. Bartlett R, Heinkenschloss M, Ridzal D, et al.. Domain decomposition methods for advection dominatedlinear-quadratic elliptic optimal control problems. Technical ReportSAND 2005-2895. Sandia National Laboratories, 2005
4. Becker R, Kapp H, Rannacher R . Adaptive finite element methods for optimal control ofpartial differential equations: basic concept. SIAM J Control Optim, 2000, 39: 113–132. doi:10.1137/S0363012999351097
5. Becker R, Vexler B . Optimal control of the convection-diffusionequation using stabilized finite element methods. Numer Math, 2007, 106(3): 349–367. doi:10.1007/s00211‐007‐0067‐0
6. Brezzi F, Russo A . Choosing bubbles for advection-diffusionproblems. Math Models Meth Appl Sci, 1994, 4: 571–587. doi:10.1142/S0218202594000327
7. Burman E, Hansbo P . Edge stabilization for Galerkinapproximations of convectiondiffusion-reaction problems. Comput Methods Appl Mech Engrg, 2004, 193: 1437–1453. doi:10.1016/j.cma.2003.12.032
8. Burman E, Hansbo P . Edge stabilization for Galerkinapproximations of the generalized Stokes' problem: A continuous interiorpenalty method. Comput Methods Appl MechEngrg, 2006, 195(19–22): 2393–2410. doi:10.1016/j.cma.2005.05.009
9. Burman E, Hansbo P . A stabilized non-conformingfinite element method for incompressible flow. Comput Methods Appl Mech Engrg, 2006, 195: 2881–2899. doi:10.1016/j.cma.2004.11.033
10. Ciarlet P G . The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland, 1978
11. Douglas Jr J, Dopont T . Interior penalty proceduresfor elliptic and parabolic Galerkin methods. In: Glowinski R, Lions J L, eds. Computing Methods in Applied Sciences. LectureNotes in Phys, Vol 58. Berlin: Springer-Verlag, 1976, 207–216
12. Fucik S, John O, Kufner A . Function Spaces. Leyden: Nordhoff, 1977
13. Fursikov A V . Optimal Control of Distributed Systems. Theory and Applications. Providence: American Mathematical Society, 2000
14. Gaevskaya A, Hoppe R H W, Repin S . A posteriori estimates for cost functionals of optimalcontrol problems. In: Bermúdez deCastro A, Gómez D, Quintela P, et al., eds. Numerical Mathematics and Advanced Applications. Berlin-Heidelberg-New York: Springer, 2006, 308–316
15. Hoppe R H W, Iliash Y, Iyyunni C, et al.. A posteriori error estimates for adaptive finiteelement discretizations of boundary control problems. J Numer Math, 2006, 14: 57–82
16. Huang Y Q, Li R, Liu W, et al.. Efficient discretization to finite element approximationof constrained optimal control problems. (to appear)
17. Hughes T J R, Rooks A . Streamline upwind/PetrovGalerkin formulations for the convection dominated flows with particularemphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Engrg, 1982, 54: 199–259
18. Johnson C . NumericalSolution of Partial Differential Equations by the Finite Element Method. Cambridge: Cambridge Univ Press, 1987
19. Johnson C, Pitkräanta J . An analysis of the discontinuousGalerkin method for scalar hyperbolic equation. Math Comp, 1986, 46: 1–26. doi:10.2307/2008211
20. Li R, Liu W, Ma H, et al.. Adaptive finite element approximation for distributedelliptic optimal control problems. SIAMJ Control Optim, 2002, 41: 1321–1349. doi:10.1137/S0363012901389342
21. Li R, Liu W, Yan N . A posteriori error estimates of recovery type for distributedconvex optimal control problems. J SciComput, 2007, 33: 155–182. doi:10.1007/s10915‐007‐9147‐7
22. Lions J L . Optimal Control of Systems Governed by Partial Differential Equations. Berlin: Springer-Verlag, 1971
23. Liu W, Ma H, Tang T, et al.. A posteriori error estimates for discontinuousGalerkin time-stepping method for optimal control problems governedby parabolic equations. SIAM Numer Anal, 2004, 42: 1032–1061. doi:10.1137/S0036142902397090
24. Liu W, Yan N . A posteriori error estimatesfor optimal boundary control. SIAM J NumerAnal, 2001, 39: 73–99. doi:10.1137/S0036142999352187
25. Liu W, Yan N . A posteriori error estimatesfor distributed convex optimal control problems. Advances in Computational Mathematics, 2001, 15: 285–309. doi:10.1023/A:1014239012739
26. Liu W, Yan N . A posteriori error estimatesfor control problems governed by stokes equations. SIAM J Numer Anal, 2002, 40: 1850–1869. doi:10.1137/S0036142901384009
27. Liu W, Yan N . A posteriori estimates foroptimal control problems governed by parabolic equations. Numer Math, 2003, 93: 497–521. doi:10.1007/s002110100380
28. Navert U . Afinite element method for convection diffusion problems. Ph D Thesis. Chalmers Inst of Tech, 1982
29. Ouazzi A, Turek S . Unified edge-oriented stabilizationof noncomforming finite element methods for incompressible flow problems. Technical Report 284. Univ at Dortmund, 2005
30. Parra-Guevara D, Skiba Y N . Elements of the mathematicalmodelling in the control of pollutants emissions. Ecological Modelling, 2003, 167: 263–275. doi:10.1016/S0304‐3800(03)00191‐1
31. Roos H G, Stynes M, Tobiska L . Numerical methods for singularly perturbed differentialequations. Ser in Computational Mathematics, Vol 24. Berlin: Springer-Verlag, 1996
32. Scott Collis S, Heinkenschloss M . Analysis of the StreamlineUpwind/Petrov Galerkin Method Applied to the Solution of Optimal ControlProblems. CAAM TR02-01, March, 2002
33. Scott L R, Zhang S . Finite element interpolationof nonsmooth functions satisfying boundary conditions. Math Comp, 1990, 54: 483–493. doi:10.2307/2008497
34. Verfürth R . Aposteriori error estimators for convection-diffusion equations. Numer Math, 1998, 80: 641–663. doi:10.1007/s002110050381
35. Yan N, Zhou Z . A priori and a posteriorierror analysis of edge stabilization Galerkin method for the optimalcontrol problem governed by convection dominated diffusion equation. Journal of Computational and Applied Mathematics (to appear)
36. Zhu J, Zeng Q . A mathematical theoreticalframe for control of air pollution. Sciencein China, Ser D, 2002, 32: 864–870