A posteriori error estimates for optimal control problems constrained by convection-diffusion equations

doi:10.1007/s11464-015-0456-0

Front. Math. China

2016, Vol. 11

Issue (1) : 55-75 https://doi.org/10.1007/s11464-015-0456-0

RESEARCH ARTICLE

A posteriori error estimates for optimal control problems constrained by convection-diffusion equations

Hongfei FU^1,^*(

),Hongxing RUI²,Zhaojie ZHOU³

1. College of Science, China University of Petroleum, Qingdao 266580, China
2. School of Mathematics, Shandong University, Jinan 250100, China
3. School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China

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Abstract

We propose a characteristic finite element discretization of evolutionary type convection-diffusion optimal control problems. Nondivergence-free velocity fields and bilateral inequality control constraints are handled. Then some residual type a posteriori error estimates are analyzed for the approximations of the control, the state, and the adjoint state. Based on the derived error estimators, we use them as error indicators in developing efficient multi-set adaptive meshes characteristic finite element algorithm for such optimal control problems. Finally, one numerical example is given to check the feasibility and validity of multi-set adaptive meshes refinements.

Keywords Optimal control problem characteristic finite element convectiondiffusion equation multi-set adaptive meshes a posterior error estimate

Corresponding Author(s): Hongfei FU

Issue Date: 02 December 2015

Cite this article:

Hongfei FU,Hongxing RUI,Zhaojie ZHOU. A posteriori error estimates for optimal control problems constrained by convection-diffusion equations[J]. Front. Math. China, 2016, 11(1): 55-75.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-015-0456-0
https://academic.hep.com.cn/fmc/EN/Y2016/V11/I1/55

1	Ainsworth M, Oden J T. A Posteriori Error Estimation in Finite Element Analysis. New York: Wiley-Interscience, 2000 https://doi.org/10.1002/9781118032824
2	Bangerth W, Rannacher R. Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics, ETH Zurich. Basel: Birkhăuser, 2003 https://doi.org/10.1007/978-3-0348-7605-6
3	Becker R, Kapp H, Rannacher R. Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J Control Optim, 2000, 39: 113–132 https://doi.org/10.1137/S0363012999351097
4	Chang Y Z, Yang D P, Zhang Z J. Adaptive finite element approximation for a class of parameter estimation problems. Appl Math Comput, 2014, 231: 284–298 https://doi.org/10.1016/j.amc.2013.12.141
5	Ciarlet P G. The Finite Element Method for Elliptic Problems. Philadelphia: SIAM, 2002 https://doi.org/10.1137/1.9780898719208
6	Douglas J, Russell T F. Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J Numer Anal, 1982, 19: 871–885DissertationTip
7	Ewing R E, Russell T F, Wheeler M F. Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Comput Methods Appl Mech Engrg, 1984, 47: 73–92 https://doi.org/10.1016/0045-7825(84)90048-3
8	Fu H F. A characteristic finite element method for optimal control problems governed by convection-diffusion equations. J Comput Appl Math, 2010, 235: 825–836 https://doi.org/10.1016/j.cam.2010.07.010
9	Fu H F, Rui H X. A priori error estimates for optimal control problems governed by transient advection-diffusion equations. J Sci Comput, 2009, 38: 290–315 https://doi.org/10.1007/s10915-008-9224-6
10	Fu H F, Rui H X. A priori and a posteriori error estimates for the method of lumped masses for parabolic optimal control problems. Int J Comput Math, 2011, 88: 2798–2823 https://doi.org/10.1080/00207160.2011.558575
11	Fu H F, Rui H X. Adaptive characteristic finite element approximation of convectiondiffusion optimal control problems. Numer Methods Partial Differential Equations, 2013, 29: 978–998 https://doi.org/10.1002/num.21741
12	Ge L, Liu W B, Yang D P. Adaptive finite element approximation for a constrained optimal control problem via multi-meshes. J Sci Comput, 2009, 41: 238–255 https://doi.org/10.1007/s10915-009-9296-y
13	Houston P, Süli E. Adaptive Lagrange-Galerkin methods for unsteady convectiondiffusion problems. Math Comput, 2001, 70: 77–106 https://doi.org/10.1090/S0025-5718-00-01187-X
14	Kufner A, John O, Fucik S. Function Spaces. Leyden: Noordhoff, 1977
15	Li R. On multi-mesh h-adaptive meshes. J Sci Comput, 2005, 24: 321–341 https://doi.org/10.1007/s10915-004-4793-5
16	Li R, Liu W B, Ma H P, Tang T. Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J Control Optim, 2002, 41: 1321–1349 https://doi.org/10.1137/S0363012901389342
17	Lions J L. Optimal Control of Systems Governed by Partial Differential Equations. Berlin: Springer-Verlag, 1971 https://doi.org/10.1007/978-3-642-65024-6
18	Liu W B, Yan N N. A posteriori error estimates for optimal boundary control. SIAM J Numer Anal, 2001, 39: 73–99 https://doi.org/10.1137/S0036142999352187
19	Liu W B, Yan N N. A posteriori error estimates for distributed convex optimal control problems. Adv Comput Math, 2001, 15: 285–309 https://doi.org/10.1023/A:1014239012739
20	Liu W B, Yan N N. A posteriori error estimates for control problems governed by nonlinear elliptic equations. Appl Numer Math, 2003, 47: 173–187 https://doi.org/10.1016/S0168-9274(03)00054-0
21	Liu W B, Yan N N. A posteriori error estimates for optimal control problems governed by parabolic equations. Numer Math, 2003, 93: 497–521 https://doi.org/10.1007/s002110100380
22	Liu W B, Yan N N. Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Beijing: Science Press, 2008
23	Meidner D, Vexler B. Adaptive space-time finite element methods for parabolic optimization problems. SIAM J Control Optim, 2007, 4: 116–142 https://doi.org/10.1137/060648994
24	Pironneau O. Optimal Shape Design for Elliptic Systems. Berlin: Springer-Verlag, 1984 https://doi.org/10.1007/978-3-642-87722-3
25	Rui H X, Tabata M. A second order characteristic finite element scheme for convectiondiffusion problems. Numer Math, 2002, 92: 161–177 https://doi.org/10.1007/s002110100364
26	Rui H X, Tabata M. A mass-conservative characteristic finite element scheme for convection-diffusion problems. J Sci Comput, 2010, 43: 416–432 https://doi.org/10.1007/s10915-009-9283-3
27	Scott L R, Zhang S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comput, 1990, 54: 483–493 https://doi.org/10.1090/S0025-5718-1990-1011446-7
28	Tiba D. Lectures on the Optimal Control of Elliptic Equations. Jyvaskyla: University of Jyvaskyla Press, 1995
29	Veeser A. Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J Numer Anal, 2001, 39: 146–167 https://doi.org/10.1137/S0036142900370812
30	Xiong C, Li Y. A posteriori error estimates for optimal distributed control governed by the evolution equations. Appl Numer Math, 2011, 61: 181–200 https://doi.org/10.1016/j.apnum.2010.09.004

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