A posteriori error estimates for convex boundary control problems

SIAM Journal on Numerical Analysis

Volumn 39, Issue 1, 2002, Pages 73-99

  Liu, Wenbin ^a   Yan, Ningning ^a,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; Optimal boundary control

Indexed keywords

EID: 0035733415     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142999352187     Document Type: Article

References (47)

1. R.A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. M. AINSWORTH, J.T. ODEN, AND C.Y. LEE, Local a posteriori error estimators for variational inequalities, Numer. Methods Partial Differential Equations, 9 (1993), pp. 22-33.
3. M. AINSWORTH AND J.T. ODEN, A posteriori error estimators in finite element analysis, Comput. Methods Appl. Mech. Engrg., 142 (1997), pp. 1-88.
4. P. ALOTTO, P. GIRDINIO, P. MOLFINO, AND M. NERVI, Mesh adaption and optimisation techniques in magnet design, IEEE Trans. Magnetics, 32 (1996).
5. W. ALT, On the approximation of infinite optimisation problems with an application to optimal control problems, Appl. Math. Optim., 12 (1984), pp. 15-27.
6. W. ALT AND U. MACKENROTH, Convergence of finite element approximations to state constrained convex parabolic boundary control problems, SIAM J. Control Optim., 27 (1989), pp. 718-736.
7. N.V. BANICHUK, F.J. BARTHOLD, A. FALK, AND E. STEIN, Mesh refinement for shape optimisation, Structural Optim., 9 (1995), pp. 45-51.
8. R.E. BANK AND A. WEISER, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 283-301.
9. J. BARANGER AND H.E. AMRI, A posteriori error estimators in finite element approximation of quasi-Newtonian flows, M2AN Math. Model. Numer. Anal., 25 (1991), pp. 31-48.
10. V. BARBU, Optimal Control for Variational Inequalities, Res. Notes Math. 100, Pitman, London, 1984.
11. J.W. BARRETT AND W.B. LIU, Finite element approximation of degenerate quasilinear elliptic and parabolic problems, in Numerical Analysis 1993, Pitman Res. Notes. Math. Ser. 303, 1994, pp. 1-16.
12. P.G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
13. Z.H. DING, L. JI, AND J.X. ZHOU, Constrained LQR problems in elliptic distributed control systems with point observations, SIAM J. Control Optim., 34 (1996), pp. 264-294.
14. R.D. DURAN, M.A. MUSCHIETTI, AND R. RODRIGUEZ, On the asymptotic exactness of error estimators for linear triangular finite elements, Numer. Math., 59 (1991), pp. 107-127.
15. F.S. FALK, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), pp. 28-47.
16. R.S. FALK, Error estimates for the approximation of a class of variational inequalities, Math. Comp., 28 (1974), pp. 963-971.
17. D. A. FRENCH AND J.T. KING, Approximation of an elliptic control problem by the finite element method, Numer. Funct. Anal. Optim., 12 (1991), pp. 299-315.
18. T. GEVECI, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Anal. Numer., 13 (1979), pp. 313-328.
19. R. GLOWINSKI, J.L. LIONS, AND R. TREMOLIERES, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1976.
20. M.D. GUNZBURGER AND L.S. HOU, Finite-dimensional approximation of a class of constrained nonlinear optimal control problems, SIAM J. Control Optim., 34 (1996), pp. 1001-1043.
21. J. HASLINGER AND P. NEITTAANMAKI, Finite Element Approximation for Optimal Shape Design, John Wiley and Sons, Chichester, UK, 1989.
22. L. HOU AND J.C. TURNER, Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls, Numer. Math., 71 (1995), pp. 289-315.
23. C. JOHNSON, Adaptive finite element methods for the obstacle problem, Math. Models Methods Appl. Sci., 2 (1992), pp. 483-487.
24. R. KORNHUBER, A posteriori error estimates for elliptic variational inequalities, Comput. Math. Appl., 31 (1996), pp. 49-60.
25. G. KNOWLES, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), pp. 414-427.
26. A. KUFNER, O. JOHN, AND S. FUCIK, Function Spaces, Nordhoff, Leiden, The Netherlands, 1977.
27. I. LASIECKA, Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions, SIAM J. Control Optim., 22 (1984), pp. 477-500.
28. J.L. LIONS, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.
29. J.L. LIONS AND E. MAGENES, Nonhomogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, 1972.
30. W.B. LIU AND J.W. BARRETT, Quasi-norm error bounds for the finite element approximation of degenerate quasilinear elliptic variational inequalities, RAIRO Anal. Numer., 28 (1994), pp. 725-744.
31. W.B. LIU AND J.W. BARRETT, Quasi-norm error bounds for the finite element approximation of degenerate quasilinear parabolic variational inequalities, Numer. Funct. Anal. Optim., 16 (1995), pp. 1309-1321.
32. W.B, LIU AND J. RUBIO, Optimality conditions for strongly monotone variational inequalities, Appl. Math. Optim., 27 (1993), pp. 291-312.
33. W.B. LIU AND D. TIBA, Error estimates in the approximation of optimization problems, Numer. Funct. Anal. Optim., to appear.
34. W.B. LIU AND N.N. YAN, A posteriori error analysis for convex distributed optimal control problems, Adv. Comput. Math., to appear.
35. K. MALANOWSKI, Convergence of approximations vs. regularity of solutions for convex, control constrained, optimal control systems, Appl. Math. Optim., 8 (1982), pp. 69-95.
36. R.S. MCKNIGHT AND W.E. BORSARGE, JR., The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim., 11 (1973), pp. 510-524.
37. R.H. NOCHETTO, Pointwise accuracy of a finite element method for nonlinear variational inequalities, Numer. Math., 54 (1989), pp. 601-618.
38. P. NEITTAANMAKI AND D. TIBA. Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, M. Dekker, New York, 1994.
39. O. PIRONNEAU, Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984.
40. L.R. SCOTT AND S. ZHANG, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483-493.
41. G. STRANG AND G. J. FIX, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.
42. D. TIBA, Optimal Control of Nonsmooth Distributed Parameter Systems, Lecture Notes Math. 1459, Springer-Verlag, Berlin, 1990.
43. D. TIBA, Lectures on the Optimal Control of Elliptic Equations, University of Jyvaskyla Press, Jyvaskyla, Finland, 1995.
44. D. TIBA AND F. TROLTZSCH, Error estimates for the discretization of state constrained convex control problems, Numer. Funct. Anal. Optim., 17 (1996), pp. 1005-1028.
45. F. TROLTZSCH, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems - strong convergence of optimal control, Appl. Math. Optim., 1994.
46. R. VERFURTH, A posteriori error estimators for the Stokes equations, Numer. Math., 55 (1989), pp. 309-325.
47. R. VERFURTH, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement, Wiley-Teubner, London, UK, 1996.