References
- F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
- H. Bium and R. Rannacher, On mixed finite element methods in plate bending analysis, Comput. Mech. 6 (1990), 221-236. https://doi.org/10.1007/BF00350239
- W. Cao and D. Yang, Ciarlet-Raviart mixed finite element approximation for an optimal control problem governed by the first bi-harmonic equation, J. Comput. Appl. Math. 233 (2009), no. 2, 372-388. https://doi.org/10.1016/j.cam.2009.07.039
-
H. Chen and Z. Jiang,
$L^{\infty}$ -convergence of mixed finite element method for laplacian operator, Korean J. Comput. Appl. Math. 7 (2000), no. 1, 61-82. - 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, Y. Huang, W. B. Liu, and N. Yan, Error estimates and superconvergence of mixed finite element methods for convex optimal control problems, J. Sci. Comput. 42 (2009), no. 3, 382-403.
- 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
- X. L. Cheng, W. M. Han, and H. C. Huang, Some mixed finite element methods for biharmonic equation, J. Comput. Appl. Math. 126 (2000), no. 1-2, 91-109. https://doi.org/10.1016/S0377-0427(99)00342-8
- 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. Duran, R. H. Nochetto, and J. Wang, Sharp maximum norm error estimates for FE approximations of the Stokes problems in 2-D, Math. Comp. 51 (1988), no. 184, 491-506.
-
J. Frehse and R. Rannacher, Eine
$L^1$ -Ferhlerabschatzung fur diskrete Grundlosungen in der Methode der Finiten Elemente, Finite Elemente (Tagung, Univ. Bonn, Bonn, 1975), pp. 92-114. Bonn. Math. Schrift., No. 89, Inst. Angew. Math., Univ. Bonn, Bonn, 1976. - C. Johnson, On the convergence of a mixed finite-element method for plate bending problems, Numer. Math. 21 (1973), 43-62. https://doi.org/10.1007/BF01436186
- B. J. Li and S. Y. Liu, On gradient-type optimization method utilizing mixed finite element approximation for optimal boundary control problem governed by bi-harmonic equation, Appl. Math. Comput. 186 (2007), no. 2, 1429-1440. https://doi.org/10.1016/j.amc.2006.07.142
- R. Li and W. Liu, http://circus.math.pku.edu.cn/AFEPack.
- J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.
- P. Monk, A mixed finite element method for the biharmonic equation, SIAM J. Numer. Anal. 24 (1987), no. 4, 737-749. https://doi.org/10.1137/0724048
- P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pp. 292-315. Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977.
- R. Scott, A Mixed method for 4th order problems using linear finite elements, RARIO Anal. Numer. 33 (1978), 681-697.
-
R. Scott, Optimal
$L^{\infty}$ -estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), no. 136, 681-697. -
J. Wang, Asympotic expansion and
$L^{\infty}$ -error estimates for mixed FEM for second order elliptic problems, Numer. Math. 55 (1989), 401-430. https://doi.org/10.1007/BF01396046 - X. Xing and Y. Chen, Error estimates of mixed finite element methods for quadratic optimal control problems, J. Comput. Appl. Math. 233 (2010), no. 8, 1812-1820. https://doi.org/10.1016/j.cam.2009.09.018
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