Special Issues

Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems

  • Received: 01 December 2019 Revised: 01 June 2020 Published: 31 July 2020
  • Primary: 49J20, 65N30; Secondary: 65N12

  • This paper aims at investigating the convergence and quasi- optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by $ L^2 $-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.

    Citation: Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems[J]. Electronic Research Archive, 2020, 28(4): 1459-1486. doi: 10.3934/era.2020077

    Related Papers:

  • This paper aims at investigating the convergence and quasi- optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by $ L^2 $-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.



    加载中


    [1] A posteriori error estimators in finite element analysis. Comput. Methods Appl. Mech. Engrg. (1997) 142: 1-88.
    [2] Error estimates for adaptive finite computations. SIAM J. Numer. Anal. (1978) 15: 736-754.
    [3] A convergent noncomforming adaptive finite element method with quasi-optimal complexity. SIAM J. Numer. Anal. (2010) 47: 4639-4659.
    [4] Adaptive finite element methods with convergence rates. Numer. Math. (2004) 97: 219-268.
    [5] Convergence analysis of a conforming adaptive finite element method for an obstacle problem. Numer. Math. (2007) 107: 455-471.
    [6] Error reduction and convergence for an adaptive mixed finite element method. Math. Comput. (2006) 75: 1033-1042.
    [7] Error estimations for the numerical approximation of semilinear elliptic control problems with finitely many state constraints. ESAIM Control Optim. Calc. Var. (2002) 8: 345-374.
    [8] Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. (2008) 46: 2524-2550.
    [9] An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math. Comput. (2004) 73: 1167-1193.
    [10] Adaptive finite element approximations for a class of nonlinear eigenvalue problems in quantum physics. Adv. Appl. Math. Mech. (2011) 3: 493-518.
    [11] Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem. Comput. Methods Appl. Mech. Engrg. (2010) 199: 1415-1423.
    [12] (2015) High Efficient and Accuracy Numerical Methods for Optimal Control Problems.Science Press.
    [13] Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer. math. (2008) 110: 313-355.
    [14] Convergence and quasi-optimality of an adaptive finite element method for controlling $L^2$ errors. Numer. Math. (2011) 117: 185-218.
    [15] A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. (1996) 33: 1106-1124.
    [16] An adaptive finite element method for linear elliptic problems. Math. Comput. (1988) 50: 361-383.
    [17] Convergence analysis of an adaptive finite element method for distributed control problems with control constrains. Inter. Ser. Numer. Math. (2007) 155: 47-68.
    [18] $L^2$ norm equivalent a posteriori error for a constraint optimal control problem. Inter. J. Numer. Anal. Model. (2009) 6: 335-353.
    [19] Adaptive finite element approximation for a constrained optimal control problem via multi-meshes. J. Sci. Comput. (2009) 41: 238-255.
    [20] Adaptive finite element method for elliptic optimal control problems: Convergence and optimality. Numer. Math. (2017) 135: 1121-1170.
    [21] W. Gong, N. Yan and Z. Zhou, Convergence of $L^2$-norm based adaptive finite element method for elliptic optimality control problem, arXiv: 1608.08699.
    [22] Convergence of adaptive finite elements for optimal control problems with control constraints. Inter. Ser. Numer. Math. (2014) 165: 403-419.
    [23] Convergence and quasi-optimality of an adaptive finite element method for optimal control problems on $L^2$-errors. J. Sci. Comput. (2017) 73: 438-458.
    [24] Convergence and quasi-optimality of an adaptive finite element method for optimal control problems with integral control constraint. Adv. Comput. Math. (2018) 44: 367-394.
    [25] Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. (2002) 41: 1321-1349.
    [26] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.
    [27] (2008) Adaptive Finite Element Methods for Optimal Control Governed by PDEs.Science Press.
    [28] A posteriori error estimates for control problems governed by nonlinear elliptic equation. Appl. Numer. Math. (2003) 47: 173-187.
    [29] Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. (2005) 43: 1803-1827.
    [30] Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. (2000) 38: 466-488.
    [31] Convergence of adaptive finite element methods. SIAM Rev. (2002) 44: 631-658.
    [32] Optimality of a standard adaptive finite element method. Found. Comput. Math. (2007) 7: 245-269.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1398) PDF downloads(200) Cited by(0)

Article outline

Figures and Tables

Figures(9)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog