|
|
|
Error estimates of triangular mixed finite element methods for quasilinear optimal control problems |
Yanping CHEN1( ), Zuliang LU2,3, Ruyi GUO4 |
| 1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China; 2. College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, China; 3. School of Mathematics and Statistics, China Three Gorges University, Chongqing 404000, China; 4. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan 411105, China |
|
|
|
|
Abstract Abstract The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial diffential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.
|
| Keywords
A priori error estimate
quasilinear elliptic equation
general convex optimal control problem
triangular mixed finite element method
|
|
Corresponding Author(s):
CHEN Yanping,Email:yanpingchen@scnu.edu.cn
|
|
Issue Date: 01 June 2012
|
|
| 1 |
Arada N, Casas E, Tr?ltzsch F. Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput Optim Appl , 2002, 23(2): 201-229
doi: 10.1023/A:1020576801966
|
| 2 |
Babuska I, Strouboulis T. The Finite Element Method and Its Reliability. Oxford: Oxford University Press, 2001
|
| 3 |
Becker R, Kapp H, Rannacher R. Adaptive ?nite element methods for optimal control of partial differential equations: basic concept. SIAM J Control Optim , 2000, 39(1): 113-132
doi: 10.1137/S0363012999351097
|
| 4 |
Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal Numer , 1974, 8(2): 129-151
|
| 5 |
Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Berlin: Springer, 1991
doi: 10.1007/978-1-4612-3172-1
|
| 6 |
Chen Y, Liu W. Posteriori error estimates for mixed ?nite elements of a quadratic optimal control problem. Recent Progress Comp Appl PDEs , 2002, 3: 123-134
doi: 10.1007/978-1-4615-0113-8_8
|
| 7 |
Chen Y, Liu W. Error estimates and superconvergence of mixed ?nite element for quadratic optimal control. Int J Num Anal Model , 2006, 3(3): 311-321
|
| 8 |
Chen Y, Liu W. A posteriori error estimates for mixed ?nite element solutions of convex optimal control problems. J Comput Appl Math , 2008, 211(1): 76-89
doi: 10.1016/j.cam.2006.11.015
|
| 9 |
Chen Y, Lu Z. Error estimates of fully discrete mixed ?nite element methods for semilinear quadratic parabolic optimal control problems. Comput Methods Appl Mech Engrg , 2010, 199(1): 1415-1423
doi: 10.1016/j.cma.2009.11.009
|
| 10 |
Chen Y, Lu Z. Error estimates for parabolic optimal control problem by fully discrete mixed ?nite element methods. Finite Elem Anal Des , 2010, 46(4): 957-965
doi: 10.1016/j.finel.2010.06.011
|
| 11 |
Falk F. Approximation of a class of optimal control problems with order of convergence estimates. J Math Anal Appl , 1973, 44(1): 28-47
doi: 10.1016/0022-247X(73)90022-X
|
| 12 |
Geveci T. On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO: Numer Anal , 1979, 33: 313-328
|
| 13 |
Grisvard P. Elliptic Problems in Nonsmooth Domains. London: Pitman, 1985
|
| 14 |
Gunzburger M, Hou S. Finite dimensional approximation of a class of constrained nonlinear control problems. SIAM J Control Optim , 1996, 34(3): 1001-1043
doi: 10.1137/S0363012994262361
|
| 15 |
Li R, Liu W. http://circus.math.pku.edu.cn/AFEPack
|
| 16 |
Li R, Liu W, Ma H, Tang T. Adaptive ?nite element approximation for distributed elliptic optimal control problems. SIAM J Control Optim , 2002, 41(5): 1321-1349
doi: 10.1137/S0363012901389342
|
| 17 |
Lions J. Optimal Control of systems Governed by Partial Differential Equations. Berlin: pringer, 1971
|
| 18 |
Liu W, Tiba D. Error estimates for the ?nite element approximation of a class of nonlinear optimal control problems. J Numer Func Optim , 2001, 22: 935-972
|
| 19 |
Liu W, Yan N. A posteriori error estimates for distributed convex optimal control problems. Adv Comput Math , 2001, 15(4): 285-309
doi: 10.1023/A:1014239012739
|
| 20 |
Liu W, Yan N. A posteriori error estimates for control problems governed by nonlinear elliptic equation. Appl Numer Math , 2003, 47(2): 173-187
doi: 10.1016/S0168-9274(03)00054-0
|
| 21 |
Lu Z, Chen Y. A posteriori error estimates of triangular mixed ?nite element methods for semilinear optimal control problems. Adv Appl Math Mech , 2009, 1(2): 242-256
|
| 22 |
Lu Z, Chen Y. L∞-error estimates of triangular mixed ?nite element methods for optimal control problem govern by semilinear elliptic equation. Numer Anal Appl , 2009, 12(1): 74-86
doi: 10.1134/S1995423909010078
|
| 23 |
Lu Z, Chen Y, Zhang H. A priori error analysis of mixed methods for nonlinear quadratic optimal control problems. Lobachevskii J Math , 2008, 29(2): 164-174
doi: 10.1134/S1995080208030074
|
| 24 |
Malanowski K. Convergence of approximation vs. regularity of solutions for convex control constrained optimal control systems. Appl Math Optim , 1981, 8(1): 69-95
doi: 10.1007/BF01447752
|
| 25 |
Miliner F. Mixed ?nite element methods for quasilinear second-order elliptic problems. Math Comp , 1985, 44(170): 303-320
doi: 10.1090/S0025-5718-1985-0777266-1
|
| 26 |
Raviart P, Thomas J. A mixed ?nite element method for 2nd order elliptic problems. In: Math Aspects of the Finite Element Method. Lecture Notes in Math, Vol 606 . Berlin: Springer, 1977, 292-315
doi: 10.1007/BFb0064470
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
| |
Shared |
|
|
|
|
| |
Discussed |
|
|
|
|
