American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games
  • DCDS-B Home
  • This Issue
  • Next Article
    The impact of toxins on competition dynamics of three species in a polluted aquatic environment
doi: 10.3934/dcdsb.2020179

A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems

Hongbo Guan 1, , Yong Yang 2, and  Huiqing Zhu 3,,

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.

Keywords: Anisotropic meshes, elliptic boundary layer equations, optimal control problems, superconvergence.
Mathematics Subject Classification: Primary: 65M60, 49K20; Secondary: 65J10.
Citation: Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020179
References:
[1]

A. Allendes, E. 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. Becker, H. 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. Gong, H. 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. Guan, D. 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. Lin, L. 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. Allendes, E. 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. Becker, H. 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. Gong, H. 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. Guan, D. 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. Lin, L. 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 Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Big element $ K_{0} $
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

$N$ 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
Table Options

Download as excel

$N$ 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
[1]

Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial & Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631
[2]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927
[3]

Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018
[4]

Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020021
[5]

Agnese Di Castro, Mayte Pérez-Llanos, José Miguel Urbano. Limits of anisotropic and degenerate elliptic problems. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1217-1229. doi: 10.3934/cpaa.2012.11.1217
[6]

Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279
[7]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[8]

Jingshi Li, Jiachuan Zhang, Guoliang Ju, Juntao You. A multi-mode expansion method for boundary optimal control problems constrained by random Poisson equations. Electronic Research Archive, 2020, 28 (2) : 977-1000. doi: 10.3934/era.2020052
[9]

Liping Wang, Chunyi Zhao. Solutions with clustered bubbles and a boundary layer of an elliptic problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2333-2357. doi: 10.3934/dcds.2014.34.2333
[10]

Liping Wang, Juncheng Wei. Solutions with interior bubble and boundary layer for an elliptic problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 333-351. doi: 10.3934/dcds.2008.21.333
[11]

Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105
[12]

Xiaowei Pang, Haiming Song, Xiaoshen Wang, Jiachuan Zhang. Efficient numerical methods for elliptic optimal control problems with random coefficient. Electronic Research Archive, 2020, 28 (2) : 1001-1022. doi: 10.3934/era.2020053
[13]

Michel Chipot, Senoussi Guesmia. On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 179-193. doi: 10.3934/cpaa.2009.8.179
[14]

Ugur G. Abdulla, Evan Cosgrove, Jonathan Goldfarb. On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems. Evolution Equations & Control Theory, 2017, 6 (3) : 319-344. doi: 10.3934/eect.2017017
[15]

Domingo Tarzia, Carolina Bollo, Claudia Gariboldi. Convergence of simultaneous distributed-boundary parabolic optimal control problems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020045
[16]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
[17]

Thomas Apel, Mariano Mateos, Johannes Pfefferer, Arnd Rösch. Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes. Mathematical Control & Related Fields, 2018, 8 (1) : 217-245. doi: 10.3934/mcrf.2018010
[18]

Gabriella Zecca. An optimal control problem for some nonlinear elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1393-1409. doi: 10.3934/dcdsb.2019021
[19]

Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331
[20]

Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673

2019 Impact Factor: 1.27

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences