A posteriori error estimates for control problems governed by nonlinear elliptic equations

Applied Numerical Mathematics

Volumn 47, Issue 2, 2003, Pages 173-187

  Liu, Wenbin ^a   Yan, Ningning ^b  

^a UNIVERSITY OF KENT   (United Kingdom)

^b INSTITUTE OF SYSTEMS SCIENCE   (China)

Author keywords

A posteriori error analysis; Adaptive finite element methods; Finite element approximation; Nonlinear optimal control

Indexed keywords

APPROXIMATION THEORY; ERROR ANALYSIS; OPTIMAL CONTROL SYSTEMS; PROBLEM SOLVING;

CONTROL PROBLEMS;

NONLINEAR EQUATIONS;

EID: 0141924668     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-9274(03)00054-0     Document Type: Conference Paper

References (28)

1. Ainsworth M., Oden J.T. A posteriori error estimators in finite element analysis. Comput. Methods Appl. Mech. Engrg. 142:1997;1-88.
2. N. Arada, E. Casas, F. Troltzsch, Error estimates for a semilinear elliptic control problem, SFB 393, Preprint 00-32, TU 2000.
3. Baranger J., Amri H.E. A posteriori error estimators in finite element approximation of quasi-Newtonian flows. Modél Math. Anal. Numér. 25:1991;31-48.
4. Becker R., Kapp H., Rannacher R. Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J. Control Optim. 39:2000;113-132.
5. Ciarlet P.G. The Finite Element Method for Elliptic Problems. 1978;North-Holland, Amsterdam.
6. Casas E., Troltzsch F. Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory. SIAM J. Optim. 13:2002;406-431.
7. Duvaut G., Lions J.L. The Inequalities in Mechanics and Physics. 1973;Springer, Berlin.
8. Falk F.S. Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44:1973;28-47.
9. Gunzburger M.D., Hou L., Svobodny Th. Analysis and finite element approximation of optimal control problems for stationary Navier-Stokes equations with Dirichlet controls. RAIRO Model. Math. Anal. Numer. 25:1991;711-748.
10. Haslinger J., Neittaanmaki P. Finite Element Approximation for Optimal Shape Design. 1989;Wiley, Chichester.
11. Ito K., Kunisch K. The augmented Lagrangian method for parameter estimation in elliptic systems. SIAM J. Control. Optim. 28:1990;113-136.
12. Kufner A., John O., Fucik S. Function Spaces. 1977;Nordhoff, Leiden.
13. Kunisch K., Tai X.C. Sequential and parallel splitting methods for bilinear control problems in Hiliber spaces. SIAM J. Numer. Anal. 34:1997;91-118.
14. Ladyzhenskaya O.A., Uraltseva N. Linear and Quasilinear Elliptic Equations. 1968;Academic Press, New York.
15. Lions J.L. Optimal Control of Systems Governed by Partial Differential Equations. 1971;Springer, Berlin.
16. Liu W.B. Recent advances in mesh adaptivity for optimal control problems. Proceedings of Fast Solution of Discretized Optimization. International Series of Numerical Mathematics (ISNM). 2001;154-166.
17. Liu W.B., Yan N.N. A posteriori error estimates for a nonlinear control problem. EUNMA '99 Proc. 2000;146-153.
18. Liu W.B., Yan N.N. Quasi-norm local error estimates for the finite element approximation of p -Laplacian. SIAM J. Numer. Anal. 39:2001;100-127.
19. Liu W.B., Yan N.N. Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p -Laplacian. Numer. Math. 89:2001;341-378.
20. Liu W.B., Yan N.N. A posteriori error estimates for optimal boundary control. SIAM J. Numer. Anal. 39:2001;73-99.
21. Liu W.B., Tiba D. Error estimates for the finite element approximation of a class of nonlinear optimal control problems. J. Numer. Funct. Optim. 22:2001;953-972.
22. Liu W.B., Yan N.N. A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math. 15:2001.
23. Malanowski K. Convergence of approximations vs. regularity of solutions for convex, control constrained, optimal control systems. Appl. Math. Optim. 8:1982.
24. Neittaanmaki P., Tiba D. Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. 1994;Dekker, New York.
25. Pironneau O. Optimal Shape Design for Elliptic Systems. 1984;Springer, Berlin.
26. Scott L.R., Zhang S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54:1990;483-493.
27. Tiba D., Troltzsch F. Error estimates for the discretization of state constrained convex control problems. Numer. Funct. Anal. Optim. 17:1996;1005-1028.
28. Verfurth R. A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement. 1996;Wiley-Teubner, Stuttgart.