Superconvergence of splitting positive definite mixed finite element for parabolic optimal control problems

References

• Chen Y, Dai Y. Superconvergence for optimal control problems governed by semi-linear elliptic equations. J Sci Comput. 2009;39:206–221.
   Web of Science ®Google Scholar
• Meyer C, Rösch A. Superconvergence properties of optimal control problems. SIAM J Control Optim. 2004;43(3):970–985.
   Web of Science ®Google Scholar
• Yang D, Chang Y, Liu W. A priori error estimates and superconvergence analysis for an optimal control problems of bilinear type. J Comput Math. 2008;4:471–487.
   Google Scholar
• Meidner D, Vexler B. A priori error estimates for space-time finite element discretization of parabolic optimal control problems part I: problems without control constraints. SIAM J Control Optim. 2008;47:1150–1177.
   Web of Science ®Google Scholar
• Meidner D, Vexler B. A priori error estimates for space-time finite element discretization of parabolic optimal control problems part II: problems with control constraints. SIAM J Control Optim. 2008;47:1301–1329.
   Web of Science ®Google Scholar
• Tang Y, Hua Y. Superconvergence of fully discrete finite element for parabolic control problems with integral constraints. East Asian J Appl Math. 2013;3(2):138–153.
   Web of Science ®Google Scholar
• Chen Y. Superconvergence of mixed finite element methods for optimal control problems. Math Comp. 2008;77:1269–1291.
   Web of Science ®Google Scholar
• Chen Y. Superconvergence of quadratic optimal control problems by triangular mixed finite elements. Int J Numer Meth Eng. 2008;75(8):881–898.
   Web of Science ®Google Scholar
• Chen Y, Huang Y, Liu W, et al. Error estimates and superconvergence of mixed finite element methods for convex optimal control problems. J Sci Comput. 2009;42(3):382–403.
   Web of Science ®Google Scholar
• Xing X, Chen Y. Superconvergence of mixed methods for optimal control problems governed by parabolic equations. Adv Appl Math Mech. 2011;3(4):401–419.
   Web of Science ®Google Scholar
• Yang D. A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media. Numer Meth Part Differ Equ. 2001;17:229–249.
   Web of Science ®Google Scholar
• Liu Y, Li H, Wang J, et al. Splitting positive definite mixed element methods for pseudo-hyperbolic equations. Numer Meth Part Differ Equ. 2012;28:670–688.
   Web of Science ®Google Scholar
• Zhang J, Yang D. A splitting positive definite mixed element method for second-order hyperbolic equations. Numer Meth Part Differ Equ. 2009;25:622–636.
   Web of Science ®Google Scholar
• Guo H, Fu H, Zhang J. A splitting positive definite mixed finite element method for elliptic optimal control problem. Appl Math Comp. 2013;219:11178–11190.
   Web of Science ®Google Scholar
• Wang F, Chen Y, Tang Y. Superconvergence of fully discrete splitting positive definite mixed FEM for hyperbolic equations. Numer Meth Part Differ Equ. 2014;30(1):175–186.
   Web of Science ®Google Scholar
• Lions J, Magenes E. Non homogeneous boundary value problems and applications. Berlin: Springer-Verlag; 1972.
   Google Scholar
• Lions J. Optimal control of systems governed by partial differential equations. Berlin: Springer-Verlag; 1971.
   Google Scholar
• Tang Y, Hua Y. Superconvergence analysis for parabolic optimal control problems. Calcolo. 2014;51:381–392.
   Web of Science ®Google Scholar
• Raviart P, Thomas J. A mixed finite element method for 2nd order elliptic problems. In: Galligani I, Magenes E, editors. Aspects of the finite element method. Vol. 606, Lecture notes in mathematics. Berlin: Springer; 1977. p. 292–315.
   Google Scholar
• Douglas J, Roberts J. Global estimates for mixed finite element methods for second order elliptic equations. Math Comp. 1985;44:39–52.
   Web of Science ®Google Scholar
• Brezzi F, Fortin M. Mixed and hybrid finite element methods. New York (NY): Springer-Verlag; 1991.
   Google Scholar
• Ewing R, Liu M, Wang J. Superconvergence of mixed finite element approximations over quadrilaterals. SIAM J Numer Anal. 1999;36:772–787.
   Web of Science ®Google Scholar
• Lin Q, Yan N. Structure and Analysis for efficient finite element methods. Baoding: Hebei University; 1996. Chinese.
   Google Scholar
• Li R, Liu W, Yan N. A posteriori error estimates of recovery type for distributed convex optimal control problems. J Sci Comput. 2002;41(5):1321–1349.
   Google Scholar
• Li R, Liu W. 2004. Available from: http://circus.math.pku.edu.cn/AFEPack.
   Google Scholar