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Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem. (English) Zbl 1231.65152
Summary: In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. By applying some error estimates techniques of mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results.

MSC:
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
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[1] Alt, W.; Machenroth, U., Convergence of finite element approximations to state constrained convex parabolic boundary control problems, SIAM J. control optim., 27, 718-736, (1989) · Zbl 0688.49032
[2] Arada, N.; Casas, E.; Tröltzsch, F., Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. optim. appl., 23, 201-229, (2002) · Zbl 1033.65044
[3] Casas, E., Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems, Adv. comput. math., 26, 137-153, (2007) · Zbl 1118.65069
[4] Chen, Y.; Liu, W.B., Posteriori error estimates for mixed finite elements of a quadratic optimal control problem, Recent prog. comput. appl. pdes, 2, 123-134, (2002) · Zbl 1066.65069
[5] Chen, Y.; Liu, W.B., Error estimates and superconvergence of mixed finite elements for quadratic optimal control, Int. J. numer. anal. model., 3, 311-321, (2006) · Zbl 1125.49026
[6] Chen, Y.; Liu, W.B., A posteriori error estimates for mixed finite element solutions of convex optimal control problems, J. comput. appl. math., 211, 76-89, (2008) · Zbl 1165.65034
[7] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[8] Evans, L., Partial differential equations, Amer. math. soc., providence, RI, (1998)
[9] Falk, F.S., Approximation of a class of optimal control problems with order of convergence estimates, J. math. anal. appl., 44, 28-47, (1973) · Zbl 0268.49036
[10] Garcia, S.M.F., Improved error estimates for mixed finite-element approximations for nonlinear parabolic equations: the continuous-time case, Numer. methods partial differen. equat., 10, 129-147, (1994) · Zbl 0792.65068
[11] Garcia, S.M.F., Improved error estimates for mixed finite-element approximations for nonlinear parabolic equations: the discrete-time case, Numer. methods partial differen. equat., 10, 149-169, (1994) · Zbl 0792.65069
[12] Gunzburger, M.D.; Hou, S.L., Finite dimensional approximation of a class of constrained nonlinear control problems, SIAM J. control optim., 34, 1001-1043, (1996) · Zbl 0849.49005
[13] Johnson, C.; Hansbo, P., Adaptive finite element methods in computational mechanics, Comput. methods appl. mech. engrg., 101, 143-181, (1992) · Zbl 0778.73071
[14] Knowles, G., Finite element approximation of parabolic time optimal control problems, SIAM J. control optim., 20, 414-427, (1982) · Zbl 0481.49026
[15] Kwon, Y.; Milner, F.A., \(L^\infty\)-error estimates for mixed methods for semilinear second-order elliptic equations, SIAM J. numer. anal., 25, 46-53, (1988) · Zbl 0643.65057
[16] R. Li, W.B. Liu, <http://circus.math.pku.edu.cn/AFEPack>.
[17] Lions, J.L., Optimal control of systems governed by partial differential equations, (1971), Springer Berlin · Zbl 0203.09001
[18] Liu, W.B.; Yan, N.N., Adaptive finite element methods for optimal control governed by pdes, (2008), Springer Beijing
[19] Lu, Z.; Chen, Y., A posteriori error estimates of triangular mixed finite element methods for semilinear optimal control problems, Adv. appl. math. mech., 1, 242-256, (2009) · Zbl 1262.49009
[20] Lu, Z.; Chen, Y., \(L^\infty\)-error estimates of triangular mixed finite element methods for optimal control problem govern by semilinear elliptic equation, Numer. anal. appl., 12, 74-86, (2009)
[21] Lu, Z.; Chen, Y.; Zhang, H., A priori error estimates of mixed finite element methods for nonlinear quadratic optimal control problems, Lobachevskii J. math., 29, 164-174, (2008) · Zbl 1172.65033
[22] Malanowski, K., Convergence of approximations vs. regularity of solutions for convex, control constrained, Appl. math. optim., 8, 69-95, (2003) · Zbl 0479.49017
[23] Meidner, D.; Vexler, B., A priori error estimates for space – time finite element discretization of parabolic optimal control problems. I. problems without control constraints, SIAM J. control optim., 47, 1150-1177, (2008) · Zbl 1161.49026
[24] Miliner, F.A., Mixed finite element methods for quasilinear second-order elliptic problems, Math. comput., 44, 303-320, (1985) · Zbl 0567.65079
[25] Neittaanmaki, P.; Tiba, D., Optimal control of nonlinear parabolic systems: theory, algorithms and applications, (1994), Marcell Dekker New York · Zbl 0812.49001
[26] Raviart, P.A.; Thomas, J.M., A mixed finite element method for 2nd order elliptic problems, Math. aspects of the finite element method, Lecture notes in math., vol. 606, (1977), Springer Berlin · Zbl 0362.65089
[27] Thomée, V., Galerkin finite element methods for parabolic problems, (1997), Springer-Verlag · Zbl 0884.65097
[28] Tiba, D.; Troltzsch, F., Error estimates for the discretization of state constrained convex control problems, Numer. funct. anal. optim., 17, 1005-1028, (1996) · Zbl 0899.49013
[29] Xiong, Z.; Chen, Y., Finite volume element method with interpolated coefficients for two-point boundary value problem of semilinear differential equations, Comput. methods appl. mech. engrg., 196, 3798-3804, (2007) · Zbl 1173.65329
[30] Xing, X.; Chen, Y., Error estimates of mixed methods for optimal control problems governed by parabolic equations, Int. J. numer. methods engrg., 75, 735-754, (2008) · Zbl 1195.65085
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