JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2016, Vol. 51 ›› Issue (2): 21-28.doi: 10.6040/j.issn.1671-9352.0.2015.190

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Iterative non-overlapping domain decomposition method for optimal boundary control problems governed by elliptic equations

LIU Wen-yue, SUN Tong-jun*   

  1. School of Mathematics, Shandong University, Jinan 250100, Shandong, China
  • Received:2015-04-23 Online:2016-02-16 Published:2016-03-11

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Abstract: A numerical method for solving optimal boundary control problems governed by elliptic equations is considered. In order to avoid large amounts of calculation produced by traditional numerical methods. An iterative non-overlapping domain decomposition method is established. The whole domain is divided into many non-overlapping subdomains, and the optimal boundary control problem is decomposed into local problems in these subdomains. Robin conditions are used to communicate the local problems on the interfaces between subdomains. The iterative scheme for solving these local problems is studied, and prove the convergence of the scheme is proved. Finally, a numerical example to prove the validity of the scheme is presented.

Key words: optimal boundary control problem, elliptic equation, Robin conditions, iterative method, non-overlapping domain decomposition method

CLC Number: 

  • O241.82
[1] LIONS J L. Optimal control of systems governed by partial differential equations[M]. New York: Springer-Verlag, 1971.
[2] NEITTAANM(¨overA)KI P, TIBA D. Optimal control of nonlinear parabolic systmes, Theroy, Algorithms and Applications[M]. Florida: CRC Press, 1994.
[3] LIU Wenbin, YAN Ningning. Adaptive finite element method for optimal control governed by PDEs[M]. Beijing: Science Press, 2008.
[4] GE Liang, LIU Wenbin, YANG Danping. Adaptive finite element approximation for a constrained optimal control problem via multi-meshes[J]. Journal of Scientific Computing, 2009, 41(2):238-255.
[5] YAN Ningning, ZHOU Zhaojie. A prior and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation[J]. Journal of Computational and Applied Mathematics, 2009, 223(1):198-217.
[6] BJORSTAD P E, WIDLUND O B. Iterative methods for the solution of elliptic problems onregionsp artitionedi nto substructures[J]. SIAM Journal on Numerical Analysis, 1986, 23(6):1097-1120.
[7] BRAMBLE J H, PASCIAK J E, SCHARTA A H. The construction of preconding for elliptic problems by substructuring[J]. Mathematics of Computation, 1986, 47(175):103-134.
[8] SUN Tongjun, MA Keying. Parapllel Galerkin domain decomposition procedures for wave euqation[J]. Journal of Computational and Applied Mathematics, 2010, 233:1850-1865.
[9] MA Keying, SUN Tongjun. Galerkin domain decomposition procedures for parabolic equations on rectangular domain[J]. International Journal for Numerical Methods in Fluids, 2010, 62(4):449-472.
[10] LOINS J L, BENSOUSSAN A, GLOWINSKI R. Méthode de décomposition appliquée au contrôle optimal de systèmes distribués[C]. 5th IFIP Conference on Optimization Techniques, Lecture Notes in Computer Science. Berlin: Springer Verlag, 1973: 5.
[11] BERGGREN M, HEINKENSCHLOSS M. Parallel solution of optimal control problems by time-domain decomposition[C] // Computational Science for the 21st Century, BRISTEAU M O. New York: Wiley, 1997.
[12] LEUGERING G. Domain decomposition of optimal control problems for dynamic networks of elastic strings[J]. Computional Optimization and Applications, 2000, 16(1):5-27.
[13] LEUGERING G. Dynamic domain decomposition of optimal control problems for networks of strings and Timoshenko beams[J]. SIAM Journal on Control and Optimal, 1999, 37(6):1649-1675.
[14] BENAMOU J D. Domain decomposition, optimal control of system governed by partial differential equations, and Sysnthesis of feedback laws[J]. Journal of Optimization Theory and Applications, 1999, 102(1):15-36.
[15] BENAMOU J D. Décompositon de domaine pour le contrôle optimal de systèmes gouvernés par des equations dEvolution[J]. Comptes Rendus de lAcadémie des Sciences de Paris, Série I, 1997, 324:1065-1070.
[16] BENAMOU J D. Domain decomposition methods with coupled transmission conditions for the optimal control of systems governed by elliptic partial differential equations[J]. SIAM Journal on Numerical Analysis, 1996, 33(6):2401-2416.
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