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Variational discretization and mixed methods for semilinear parabolic optimal control problems with integral constraint | ||
| Journal of Computational & Applied Research in Mechanical Engineering (JCARME) | ||
| مقاله 3، دوره 1، شماره 1، اسفند 2011، صفحه 29-36 اصل مقاله (437.15 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22061/jcarme.2011.1 | ||
| نویسنده | ||
| Zuliang Lu* | ||
| School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000, P.R.China; College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, P.R.China | ||
| تاریخ دریافت: 09 تیر 1390، تاریخ بازنگری: 03 شهریور 1390، تاریخ پذیرش: 07 شهریور 1390 | ||
| چکیده | ||
| The aim of this work is to investigate the variational discretization and mixed finite element methods for optimal control problem governed by semi linear parabolic equations with integral constraint. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is not discreted. Optimal error estimates in L2 are established for the state and the control variable. As a result, it can be proved that the discrete solutions possess the convergence property of order. Finally, a numerical example is presented which confirms the theoretical results. | ||
| کلیدواژه‌ها | ||
| Priori error estimates؛ Parabolic optimal control؛ Integral constraint؛ Mixed finite element method؛ Variational discretization | ||
| مراجع | ||
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[1] F. S. Falk, “Approximation of a class of optimal control problems with order of convergence estimates”, J. Math. Anal. Appl., Vol. 44, pp. 28-47, (1973). [2] T. Geveci, “In the approximation of the solution of an optimal control problem governed by an elliptic equations”, R.A.I.R.O. Numer. Anal., Vol. 13, pp. 313-328, (1979). [3] W. B. Liu and N. N. Yan, “Adaptive finite element methods for optimal control governed by PDEs”, Science Press, Beijing, (2008). [4] Y. Chen, "A posteriori error estimates for mixed finite element solutions of convex optimal control problems ," J. Comp. Appl. Math., Vol. 211, pp. 76-89, (2008). [5] Z. Lu and Y. Chen, “A posteriori error estimates of triangular mixed finite element methods for semi-linear optimal control problems”, Adv. Appl. Math. Mech., Vol. 1, pp. 242-256, (2009). [6] Z. Lu and Y. Chen, “-error estimates of triangular mixed finite element methods for optimal control problem govern by semilinear elliptic equation”, Numer. Anal. Appl., Vol. 12, pp. 74-86, (2009). [7] Y. Chen and Z. Lu, “Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problems”, Comput. Methods Appl. Mech. Eng., Vol. 199, pp. 1415-1423, (2010). [8] M. Hinze, “A variational discretization concept in control constrained optimization: the linear quadratic case”, J. Comput. Optim. Appl., Vol. 30, pp. 45-63, (2010). [9] Z. Lu, Y. Chen and H. Zhang, “A priori error analysis of mixed methods for nonlinear quadratic optimal control problem”, Lobachevskii J. Math., Vol. 29, pp. 164-174, (2007). [10] Y. Chen and Z. Lu, “Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods”, Finite Elem. Anal. Des., Vol. 46, pp. 957-965, (2010). [11] Y. Chen, L. Dai and Z. Lu, “Superconvergence of rectangular mixed finite element methods for constrained optimal control problem”, Adv. Appl. Math. Mech, Vol. 2, pp.56-75, (2010). [12] Y. Chen and W. B. Liu, “Error estimates and superconvergence of mixed finite elements for quadratic optimal control,” Internat. J. Numer. Anal. Modeling, Vol. 3, pp. 311-321, (2006). | ||
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