References
- W. Alt, On the approximation of infinite optimization problems with an application to optimal control problems, Appl. Math. Optim. 12 (1984), no. 1, 15-27. https://doi.org/10.1007/BF01449031
- W. Alt and U. Machenroth, Convergence of finite element approximations to state constrained convex parabolic boundary control problems, SIAM J. Control Optim. 27 (1989), no. 4, 718-736. https://doi.org/10.1137/0327038
- N. Arada, E. Casas, and F. Troltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl. 23 (2002), no. 2, 201- 229. https://doi.org/10.1023/A:1020576801966
- H. Blum, Q. Lin, and R. Rannacher, Asymptotic error expansion and Richardson ex- trapolation for linear finite elements, Numer. Math. 49 (1986), no. 1, 11-37. https://doi.org/10.1007/BF01389427
- H. Brunner, Y. Lin, and S. Zhang, Higher accuracy methods for second-kind Volterra in- tegral equations based on asymptotic expansions of iterated Galerkin methods, J. Integral Equations Appl. 10 (1998), no. 4, 375-396. https://doi.org/10.1216/jiea/1181074245
- C. Chen and Y. Huang, Higher Accuracy Theory of FEM, Hunan Science Press, Changsha, China, 1995.
- Y. Chen, Superconvergence of mixed finite element methods for optimal control problems, Math. Comp. 77 (2008), no. 263, 1269-1291. https://doi.org/10.1090/S0025-5718-08-02104-2
- Y. Chen, Superconvergence of quadratic optimal control problems by triangular mixed finite elements, Internat. J. Numer. Methods Engrg. 75 (2008), no. 8, 881-898. https://doi.org/10.1002/nme.2272
- Y. Chen and W. B. Liu, Error estimates and superconvergence of mixed finite elements for quadratic optimal control, Int. J. Numer. Anal. Model. 3 (2006), no. 3, 311-321.
- Y. Chen and W. B. Liu, A posteriori error estimates for mixed finite element solutions of convex optimal control problems, J. Comput. Appl. Math. 211 (2008), no. 1, 76-89. https://doi.org/10.1016/j.cam.2006.11.015
- Y. Chen and Z. Lu, Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 23-24, 1415-1423. https://doi.org/10.1016/j.cma.2009.11.009
- I. Chryssoverghi, Discretization methods for semilinear parabolic optimal control problems, Int. J. Numer. Anal. Model. 3 (2006), no. 4, 437-458.
- J. Douglas and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39-52. https://doi.org/10.1090/S0025-5718-1985-0771029-9
-
R. E. Ewing, Y. Lin, T. Sun, J. Wang, and S. Zhang, Sharp
$L^{2}$ -error estimates and superconvergence of mixed finite element methods for non-Fickian ows in porous media, SIAM J. Numer. Anal. 40 (2002), no. 4, 1538-1560. https://doi.org/10.1137/S0036142900378406 - G. Fairweather, Q. Lin, Y. Lin, J. Wang, and S. Zhang, Asymptotic expansions and Richardson extrapolation of approximate solutions for second order elliptic problems on rectangular domains by mixed finite element methods, SIAM J. Numer. Anal. 44 (2006), no. 3, 1122-1149. https://doi.org/10.1137/040614293
- M. D. Gunzburger and S. L. Hou, Finite-dimensional approximation of a class of constrained nonlinear optimal control problems, SIAM J. Control Optim. 34 (1996), no. 3, 1001-1043. https://doi.org/10.1137/S0363012994262361
- J. Haslinger and P. Neittaanmaki, Finite Element Approximation for Optimal Shape Design, John Wiley and Sons, Chichester, UK, 1988.
- P. Helfrich, Asymptotic expansion for the finite element approximation of parabolic problems, Extrapolation procedures in the finite element method (Bonn, 1983), 11-30, Bonner Math. Schriften, 158, Univ. Bonn, Bonn, 1984.
- 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), no. 3, 289-315. https://doi.org/10.1007/s002110050146
- G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim. 20 (1982), no. 3, 414-427. https://doi.org/10.1137/0320032
- Q. Lin, I. H. Sloan, and R. Xie, Extrapolation of the iterated-collocation method for integral equations, SIAM J. Numer. Anal. 27 (1990), no. 6, 1535-1541. https://doi.org/10.1137/0727090
- Q. Lin and N. Yan, The Construction and Analysis of High Efficiency Finite Element Methods, Hebei University Press, Baoding, China, 1996.
- Q. Lin, S. Zhang, and N. Yan, Extrapolation and defect correction for diffusion equations with boundary integral conditions, Acta Math. Sci. Ser. B Engl. Ed. 17 (1997), no. 4, 405-412.
- Q. Lin, S. Zhang, and N. Yan, High accuracy analysis for integrodifferential equations, Acta Math. Appl. Sin. Engl. Ser. 14 (1998), no. 2, 202-211. https://doi.org/10.1007/BF02677428
- Q. Lin, S. Zhang, and N. Yan, Methods for improving approximate accuracy for hyperbolic integrodifferential equations, J. Systems Sci. Math. Sci. 10 (1997), no. 3, 282-288.
- T. Lin, Y. Lin, M. Rao, and S. Zhang, Petrov-Galerkin methods for linear Volterra integro-differential equations, SIAM J. Numer. Anal. 38 (2000), no. 3, 937-963. https://doi.org/10.1137/S0036142999336145
- J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.
- H. Liu and N. Yan, Global Superconvergence for optimal control problems governed by Stokes equations, Int. J. Numer. Anal. Model. 3 (2006), no. 3, 283-302.
- T. Liu, N. Yan, and S. Zhang, Richardson extrapolation and defect correction of finite element methods for optimal control problems, J. Comput. Math. 28 (2010), no. 1, 55-71.
- R. S. Mcknight and W. E. Borsarge, The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim. 11 (1973), 510-524. https://doi.org/10.1137/0311040
- P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, M. Dekker, New York, 1994.
- D. Tiba, Lectures on the Optimal Control of Elliptic Problems, University of Jyvaskyla Press, Jyvaskyla, Finland, 1995.
- F. Troltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal control, Appl. Math. Optim. 29 (1994), no. 3, 309-329. https://doi.org/10.1007/BF01189480
- J. Wang, Superconvergence and extrapolation for mixed finite element methods on rectangular domains, Math. Comp. 56 (1991), no. 194, 477-503. https://doi.org/10.1090/S0025-5718-1991-1068807-0
-
J. Wang, Asymptotic expansions and
$L^{\infty}$ -error estimates for mixed finite element meth- ods for second order elliptic problems, Numer. Math. 55 (1989), no. 4, 401-430. https://doi.org/10.1007/BF01396046 - X. Xing and Y. Chen, Error estimates of mixed methods for optimal control problems governed by parabolic equations, Internat. J. Numer. Methods Engrg. 75 (2008), no. 6, 735-754. https://doi.org/10.1002/nme.2289
- N. Yan and K. Li, An extrapolation method for BEM, J. Comput. Math. 2 (1989), no. 2, 217-224.
- S. Zhang, T. Lin, Y. Lin, and M. Rao, Extrapolation and a-posteriori error estimators of Petrov-Galerkin methods for non-linear Volterra integro-differential equations, J. Comput. Math. 19 (2001), no. 4, 407-422.
Cited by
- The Ellis semigroup of a nonautonomous discrete dynamical system 2017, https://doi.org/10.2989/16073606.2017.1313332