March  2021, 26(3): 1711-1722. doi: 10.3934/dcdsb.2020179

A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems

1. 

College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China

2. 

College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

3. 

School of Mathematics and Natural Sciences, The University of Southern Mississippi, Hattiesburg, MS 39406, USA

* Corresponding author: Huiqing Zhu

Received  May 2019 Revised  February 2020 Published  June 2020

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

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.

Citation: Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179
References:
[1]

A. AllendesE. Hernández and E. Otárola, A robust numerical method for a control problem involving singularly perturbed equations, Computers and Mathematics with Applications, 72 (2016), 974-991.  doi: 10.1016/j.camwa.2016.06.010.  Google Scholar

[2]

T. Apel and G. Lube, Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem, Applied Numerical Mathematics, 26 (1998), 415-433.  doi: 10.1016/S0168-9274(97)00106-2.  Google Scholar

[3]

R. BeckerH. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM Journal on Control and Optimization, 39 (2000), 113-132.  doi: 10.1137/S0363012999351097.  Google Scholar

[4]

J. Bonnans and H. Zidani, Optimal control problems with partially polyhedric constraints, SIAM Journal on Control and Optimization, 37 (1999), 1726-1741.  doi: 10.1137/S0363012998333724.  Google Scholar

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S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-1-4757-4338-8.  Google Scholar

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P. Das, An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh, Numerical Algorithms, 81 (2019), 465-487.  doi: 10.1007/s11075-018-0557-4.  Google Scholar

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R. Dur$\acute{a}$n and A. Lombardi, Error estimates on anisotropic $Q_1$ elements for functions in weighted Sobolev spaces, Mathematics of Computation, 74 (2005), 1679-1706.  doi: 10.1090/S0025-5718-05-01732-1.  Google Scholar

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R. S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, Journal of Mathematical Analysis and Applications, 44 (1973), 28-47.  doi: 10.1016/0022-247X(73)90022-X.  Google Scholar

[9]

W. Gong and N. N. Yan, Adaptive finite element method for elliptic optimal control problems: Convergence and optimality, Numerische Mathematik, 135 (2017), 1121-1170.  doi: 10.1007/s00211-016-0827-9.  Google Scholar

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W. GongH. P. Liu and N. N. Yan, Adaptive finite element method for parabolic equations with Dirac measure, Computer Methods in Applied Mechanics and Engineering, 328 (2018), 217-241.  doi: 10.1016/j.cma.2017.08.051.  Google Scholar

[11]

H. B. Guan and D. Y. Shi, A high accuracy NFEM for constrained optimal control problems governed by elliptic equations, Applied Mathematics and Computation, 245 (2014), 382-390.  doi: 10.1016/j.amc.2014.07.077.  Google Scholar

[12]

H. B. GuanD. Y. Shi and X. F. Guan, High accuracy analysis of nonconforming MFEM for constrained optimal control problems governed by Stokes equations, Applied Mathematics Letters, 53 (2016), 17-24.  doi: 10.1016/j.aml.2015.09.016.  Google Scholar

[13]

H. B. Guan and D. Y. Shi, An efficient NFEM for optimal control problems governed by a bilinear state equation, Computers and Mathematics with Applications, 77 (2019), 1821-1827.  doi: 10.1016/j.camwa.2018.11.017.  Google Scholar

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W. Hackbusch, Multigrid Methods and Applications, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-02427-0.  Google Scholar

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M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications, 30 (2005), 45-63.  doi: 10.1007/s10589-005-4559-5.  Google Scholar

[16]

S. Kumar and M. Kumar, An analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems, Journal of Computational and Applied Mathematics, 281 (2015), 250-262.  doi: 10.1016/j.cam.2014.12.018.  Google Scholar

[17]

S. Kumar and S. C. S. Rao, A robust domain decomposition algorithm for singularly perturbed semilinear systems, International Journal of Computer Mathematics, 94 (2017), 1108-1122.  doi: 10.1080/00207160.2016.1184257.  Google Scholar

[18]

J. C. Li, Convergence and superconvergence analysis of finite element methods on highly nonuniform anisotropic meshes for singularly perturbed reaction-diffusion problems, Applied Numerical Mathematics, 36 (2001), 129-154.  doi: 10.1016/S0168-9274(99)00145-2.  Google Scholar

[19]

J. C. Li and M. F. Wheeler, Uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids, SIAM Journal on Numerical Analysis, 38 (2000), 770-798.  doi: 10.1137/S0036142999351212.  Google Scholar

[20]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.  Google Scholar

[21]

Q. LinL. Tobiska and A.H. Zhou, Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation, IMA Journal of Numerical Analysis, 25 (2005), 160-181.  doi: 10.1093/imanum/drh008.  Google Scholar

[22]

L. B. Liu and Y. P. Chen, An adaptive moving grid method for a system of singularly perturbed initial value problems, Journal of Computational and Applied Mathematics, 274 (2015), 11-22.  doi: 10.1016/j.cam.2014.06.022.  Google Scholar

[23]

W. B. Liu and N. N. Yan, A posteriori error estimates for control problems governed by Stokes equations, SIAM Journal on Numerical Analysis, 40 (2002), 1850-1869.  doi: 10.1137/S0036142901384009.  Google Scholar

[24]

G. Lube and B. Tews, Optimal control of singularly perturbed advection-diffusion-reaction problems, Mathematical Models and Methods in Applied Sciences, 20 (2010), 375-395.  doi: 10.1142/S0218202510004271.  Google Scholar

[25]

J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1995. doi: 10.1142/2933.  Google Scholar

[26]

H.-G. Roos, Layer-adapted grids for singular perturbation problems, Journal of Applied Mathematics and Mechanics, 78 (1998), 291-309.  doi: 10.1002/(SICI)1521-4001(199805)78:5<291::AID-ZAMM291>3.0.CO;2-R.  Google Scholar

[27]

H.-G. Roos and C. Reibiger, Numerical analysis of a system of singularly perturbed convection-diffusion equations related to optimal control, Numerical Mathematics: Theory, Methods and Applications, 4 (2011), 562-575.  doi: 10.4208/nmtma.2011.m1101.  Google Scholar

[28]

Z. M. Zhang, Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems, Mathematics of Computation, 72 (2003), 1147-1177.  doi: 10.1090/S0025-5718-03-01486-8.  Google Scholar

[29]

Z. M. Zhang and H. Q. Zhu, Uniform convergence of the LDG method for a singularly perturbed problem with the exponential boundary layer, Mathematics of Computation, 83 (2014), 635-663.  doi: 10.1090/S0025-5718-2013-02736-6.  Google Scholar

[30]

H. Q. Zhu and Z. M. Zhang, Convergence analysis of the LDG method applied to singularly perturbed problems, Numerical Methods for Partial Differential Equations, 29 (2013), 396-421.  doi: 10.1002/num.21711.  Google Scholar

show all references

References:
[1]

A. AllendesE. Hernández and E. Otárola, A robust numerical method for a control problem involving singularly perturbed equations, Computers and Mathematics with Applications, 72 (2016), 974-991.  doi: 10.1016/j.camwa.2016.06.010.  Google Scholar

[2]

T. Apel and G. Lube, Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem, Applied Numerical Mathematics, 26 (1998), 415-433.  doi: 10.1016/S0168-9274(97)00106-2.  Google Scholar

[3]

R. BeckerH. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM Journal on Control and Optimization, 39 (2000), 113-132.  doi: 10.1137/S0363012999351097.  Google Scholar

[4]

J. Bonnans and H. Zidani, Optimal control problems with partially polyhedric constraints, SIAM Journal on Control and Optimization, 37 (1999), 1726-1741.  doi: 10.1137/S0363012998333724.  Google Scholar

[5]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-1-4757-4338-8.  Google Scholar

[6]

P. Das, An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh, Numerical Algorithms, 81 (2019), 465-487.  doi: 10.1007/s11075-018-0557-4.  Google Scholar

[7]

R. Dur$\acute{a}$n and A. Lombardi, Error estimates on anisotropic $Q_1$ elements for functions in weighted Sobolev spaces, Mathematics of Computation, 74 (2005), 1679-1706.  doi: 10.1090/S0025-5718-05-01732-1.  Google Scholar

[8]

R. S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, Journal of Mathematical Analysis and Applications, 44 (1973), 28-47.  doi: 10.1016/0022-247X(73)90022-X.  Google Scholar

[9]

W. Gong and N. N. Yan, Adaptive finite element method for elliptic optimal control problems: Convergence and optimality, Numerische Mathematik, 135 (2017), 1121-1170.  doi: 10.1007/s00211-016-0827-9.  Google Scholar

[10]

W. GongH. P. Liu and N. N. Yan, Adaptive finite element method for parabolic equations with Dirac measure, Computer Methods in Applied Mechanics and Engineering, 328 (2018), 217-241.  doi: 10.1016/j.cma.2017.08.051.  Google Scholar

[11]

H. B. Guan and D. Y. Shi, A high accuracy NFEM for constrained optimal control problems governed by elliptic equations, Applied Mathematics and Computation, 245 (2014), 382-390.  doi: 10.1016/j.amc.2014.07.077.  Google Scholar

[12]

H. B. GuanD. Y. Shi and X. F. Guan, High accuracy analysis of nonconforming MFEM for constrained optimal control problems governed by Stokes equations, Applied Mathematics Letters, 53 (2016), 17-24.  doi: 10.1016/j.aml.2015.09.016.  Google Scholar

[13]

H. B. Guan and D. Y. Shi, An efficient NFEM for optimal control problems governed by a bilinear state equation, Computers and Mathematics with Applications, 77 (2019), 1821-1827.  doi: 10.1016/j.camwa.2018.11.017.  Google Scholar

[14]

W. Hackbusch, Multigrid Methods and Applications, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-02427-0.  Google Scholar

[15]

M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications, 30 (2005), 45-63.  doi: 10.1007/s10589-005-4559-5.  Google Scholar

[16]

S. Kumar and M. Kumar, An analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems, Journal of Computational and Applied Mathematics, 281 (2015), 250-262.  doi: 10.1016/j.cam.2014.12.018.  Google Scholar

[17]

S. Kumar and S. C. S. Rao, A robust domain decomposition algorithm for singularly perturbed semilinear systems, International Journal of Computer Mathematics, 94 (2017), 1108-1122.  doi: 10.1080/00207160.2016.1184257.  Google Scholar

[18]

J. C. Li, Convergence and superconvergence analysis of finite element methods on highly nonuniform anisotropic meshes for singularly perturbed reaction-diffusion problems, Applied Numerical Mathematics, 36 (2001), 129-154.  doi: 10.1016/S0168-9274(99)00145-2.  Google Scholar

[19]

J. C. Li and M. F. Wheeler, Uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids, SIAM Journal on Numerical Analysis, 38 (2000), 770-798.  doi: 10.1137/S0036142999351212.  Google Scholar

[20]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.  Google Scholar

[21]

Q. LinL. Tobiska and A.H. Zhou, Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation, IMA Journal of Numerical Analysis, 25 (2005), 160-181.  doi: 10.1093/imanum/drh008.  Google Scholar

[22]

L. B. Liu and Y. P. Chen, An adaptive moving grid method for a system of singularly perturbed initial value problems, Journal of Computational and Applied Mathematics, 274 (2015), 11-22.  doi: 10.1016/j.cam.2014.06.022.  Google Scholar

[23]

W. B. Liu and N. N. Yan, A posteriori error estimates for control problems governed by Stokes equations, SIAM Journal on Numerical Analysis, 40 (2002), 1850-1869.  doi: 10.1137/S0036142901384009.  Google Scholar

[24]

G. Lube and B. Tews, Optimal control of singularly perturbed advection-diffusion-reaction problems, Mathematical Models and Methods in Applied Sciences, 20 (2010), 375-395.  doi: 10.1142/S0218202510004271.  Google Scholar

[25]

J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1995. doi: 10.1142/2933.  Google Scholar

[26]

H.-G. Roos, Layer-adapted grids for singular perturbation problems, Journal of Applied Mathematics and Mechanics, 78 (1998), 291-309.  doi: 10.1002/(SICI)1521-4001(199805)78:5<291::AID-ZAMM291>3.0.CO;2-R.  Google Scholar

[27]

H.-G. Roos and C. Reibiger, Numerical analysis of a system of singularly perturbed convection-diffusion equations related to optimal control, Numerical Mathematics: Theory, Methods and Applications, 4 (2011), 562-575.  doi: 10.4208/nmtma.2011.m1101.  Google Scholar

[28]

Z. M. Zhang, Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems, Mathematics of Computation, 72 (2003), 1147-1177.  doi: 10.1090/S0025-5718-03-01486-8.  Google Scholar

[29]

Z. M. Zhang and H. Q. Zhu, Uniform convergence of the LDG method for a singularly perturbed problem with the exponential boundary layer, Mathematics of Computation, 83 (2014), 635-663.  doi: 10.1090/S0025-5718-2013-02736-6.  Google Scholar

[30]

H. Q. Zhu and Z. M. Zhang, Convergence analysis of the LDG method applied to singularly perturbed problems, Numerical Methods for Partial Differential Equations, 29 (2013), 396-421.  doi: 10.1002/num.21711.  Google Scholar

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
$N$ 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
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
$N$ 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
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