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椭圆最优控制问题分裂正定混合有限元方法的超收敛性分析

唐跃龙, 华玉春   

  1. 湖南科技学院, 永州 425199
  • 收稿日期:2020-06-30 出版日期:2021-11-14 发布日期:2021-11-12
  • 基金资助:
    国家自然科学基金项目(11401201),湖南省自然科学基金项目(2020JJ4323),湖南省教育厅科学研究项目(20A211,20C0854),湖南科技学院科学研究项目(18XKY063,20XKY059),湖南科技学院应用特色学科建设项目资助.

唐跃龙, 华玉春. 椭圆最优控制问题分裂正定混合有限元方法的超收敛性分析[J]. 计算数学, 2021, 43(4): 506-515.

Tang Yuelong, Hua Yuchun. SUPERCONVERGENCE ANALYSIS OF SPLITTING POSITIVE DEFINITE MIXED FINITE ELEMENT FOR ELLIPTIC OPTIMAL CONTROL PROBLEM[J]. Mathematica Numerica Sinica, 2021, 43(4): 506-515.

SUPERCONVERGENCE ANALYSIS OF SPLITTING POSITIVE DEFINITE MIXED FINITE ELEMENT FOR ELLIPTIC OPTIMAL CONTROL PROBLEM

Tang Yuelong, Hua Yuchun   

  1. Hunan University of Science and Engineering, Yongzhou 425199, China
  • Received:2020-06-30 Online:2021-11-14 Published:2021-11-12
首先利用变分原理和最优化理论得到了原问题的等价最优性条件;其次构造了椭圆最优控制问题分裂正定混合有限元方法的逼近格式;再次通过引入一些重要的中间变量和投影算子,并利用投影算子的相关性质,结合分裂正定混合有限元本身的逼近结果,得到了椭圆最优控制问题分裂正定混合有限元方法的超收敛性;最后数值实验结果验证了所得理论结果的正确性.
Firstly, we use variational principle and optimization theory to obtain the equivalent optimality conditions of the original problem. Secondly, we construct a splitting positive definite mixed finite element approximation scheme for the elliptic optimal control problem. Thirdly, by introducing some important intermediate variables and projection operator and utilizing their properties, we derive the superconvergence based on the approximation properties of splitting positive definite mixed finite element method. Finally, a numerical example is provided to validate our theoretical results.

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[1] Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods[M]. Berlin:Springer-Verlag, 1991.
[2] Chen Y, Dai Y. Superconvergence for optimal control problems governed by semi-linear elliptic equations[J]. J. Sci. Comput., 2009, 39:206-221.
[3] Chen Y, Lu Z, Huang Y. Superconvergence of triangular Raviart-Thomas mixed finite element methods for bilinear constrained optimal control problem[J]. Comput. Math. Appl., 2013, 66(8):1498-1513.
[4] Douglas J, Roberts J. Global estimates for mixed finite element methods for second order elliptic equations[J]. Math. Comp., 1985, 44:39-52.
[5] Ewing R, Liu M, Wang J. Superconvergence of mixed finite element approximations over quadrilaterals[J]. SIAM J. Numer. Anal., 1999, 36:772-787.
[6] Guo H, Fu H, Zhang J. A splitting positive definite mixed finite element method for elliptic optimal control problem[J]. Appl. Math. Comput., 2013, 219:11178-11190.
[7] Hinze M. A variational discretization concept in control constrained optimization:the linearquadratic case[J]. Comput. Optim. Appl., 2005, 30, 45-63.
[8] Li R, Liu W, Yan N. A posteriori error estimates of recovery type for distributed convex optimal control problems[J]. J. Sci. Comput., 2002, 41(5):1321-1349.
[9] 林群, 严宁宁. 高效有限元构造与分析[M]. 保定, 河北大学出版社, 1996.
[10] Lions J. Optimal Control of Systems Governed by Partial Differential Equations[M]. Berlin:Springer-Verlag, 1971.
[11] Liu W, Yan N. Adaptive Finite Element Methods for Optimal Control Governed by PDEs[M]. Beijing:Science Press, 2008.
[12] Tang Y, Hua Y. Superconvergence of splitting positive definite mixed finite element for parabolic optimal control problems[J]. Anal. Appl., 2018, 97(16):2778-2793.
[13] Wang F, Chen Y, Tang Y. Superconvergence of fully discrete splitting positive definite mixed FEM for hyperbolic equations[J]. Numer. Meth. Part. Differ. Equ., 2014, 30(1):175-186.
[14] Yang D. A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media[J]. Numer. Meth. Part. Differ. Equ., 2001, 17:229-249.
[15] Zhang J, Yang D. A splitting positive definite mixed element method for second-order hyperbolic equations[J]. Numer. Meth. Part. Differ. Equ., 2008, 25(3):622-636.
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