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A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems

  • * Corresponding author: Huiqing Zhu

    * Corresponding author: Huiqing Zhu

The first author is supported by National Natural Science Foundation of China (No. 11501527)

Abstract Full Text(HTML) Figure(6) / Table(2) Related Papers Cited by
  • In this paper, an anisotropic bilinear finite element method is constructed for the elliptic boundary layer optimal control problems. Supercloseness properties of the numerical state and numerical adjoint state in a $ \epsilon $-norm are established on anisotropic meshes. Moreover, an interpolation type post-processed solution is shown to be superconvergent of order $ O(N^{-2}) $, where the total number of nodes is of $ O(N^2) $. Finally, numerical results are provided to verify the theoretical analysis.

    Mathematics Subject Classification: Primary: 65M60, 49K20; Secondary: 65J10.


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  • Figure 1.  Anisotropic mesh (left) and uniform mesh (right)

    Figure 2.  Big element $ K_{0} $

    Figure 3.  The profile of $ y_h $ (left plot) and $ p_h $ (right plot) on a uniform mesh with $ N = 8 $

    Figure 4.  Pointwise errors of $ |y-y_h| $ (left plot) and $ |p-p_h| $ (right plot) on a uniform mesh with $ N = 8 $

    Figure 5.  The profile of $ y_h $ (left plot) and $ p_h $ (right plot) on an anisotropic mesh with $ N = 8 $

    Figure 6.  Pointwise errors of $ |y-y_h| $ (left plot) and $ |p-p_h| $ (right plot) on an anisotropic mesh with $ N = 8 $

    Table 1.  Errors and convergence rates on uniform meshes

    $N$ 4 8 16 32 64
    $\|u-u_h\|_{0}$ 4.1470E-02 2.7610E-02 1.7622E-02 1.0661E-02 5.6118E-03
    order / 0.5869 0.6478 0.7250 0.9258
    $|||\Pi_{h} y-y_h|||$ 5.0583E-02 3.5215E-02 2.4487E-02 1.6591E-02 9.9351E-03
    order / 0.5225 0.5242 0.5616 0.7398
    $|||\Pi_{h} p-p_h|||$ 4.8403E-02 3.4837E-02 2.4420E-02 1.6569E-02 9.9076E-03
    order / 0.4745 0.5126 0.5596 0.7419
    $|||y-\Pi_{2h}y_h|||$ 1.7466E-01 9.7410E-02 5.7862E-02 3.5868E-02 2.1591E-02
    order / 0.8424 0.7515 0.6899 0.7323
    $|||p-\Pi_{2h}p_h|||$ 1.7207E-01 9.6955E-02 5.7777E-02 3.5844E-02 2.1562E-02
    order / 0.8276 0.7468 0.6888 0.7332
     | Show Table
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    Table 2.  Errors and convergence rates on anisotropic meshes

    $N$ 4 8 16 32 64
    $\|u-u_h\|_{0}$ 1.1762E-02 3.0109E-03 7.5308E-04 1.8747E-04 4.6824E-05
    order / 1.9659 1.9993 2.0062 2.0013
    $|||\Pi_{h} y-y_h|||$ 8.5488E-03 2.9582E-03 8.8988E-04 2.3807E-04 6.2920E-05
    order / 1.5310 1.7330 1.9022 1.9198
    $|||\Pi_{h} p-p_h|||$ 6.3618E-03 2.5354E-03 7.8790E-04 2.1187E-04 5.8477E-05
    order / 1.3272 1.6861 1.8949 1.8572
    $|||y-\Pi_{2h}y_h|||$ 1.2562E-02 4.3649E-03 1.2406E-03 3.0455E-04 7.6278E-05
    order / 1.5250 1.8149 2.0263 1.9974
    $|||p-\Pi_{2h}p_h|||$ 1.0955E-02 4.0558E-03 1.1598E-03 2.8267E-04 7.3036E-05
    order / 1.4335 1.8061 2.0367 1.9524
     | Show Table
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