
-
Previous Article
Existence results for fractional differential equations in presence of upper and lower solutions
- DCDS-B Home
- This Issue
-
Next Article
Finite element approximation of nonlocal dynamic fracture models
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 |
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) $. Finally, numerical results are provided to verify the theoretical analysis.
References:
[1] |
A. Allendes, E. Hernández and E. Otárola,
A robust numerical method for a control problem involving singularly perturbed equations, Computers and Mathematics with Applications, 72 (2016), 974-991.
doi: 10.1016/j.camwa.2016.06.010. |
[2] |
T. Apel and G. Lube,
Anisotropic mesh refinement for a singularly perturbed reaction diffusion model problem, Applied Numerical Mathematics, 26 (1998), 415-433.
doi: 10.1016/S0168-9274(97)00106-2. |
[3] |
R. Becker, H. Kapp and R. Rannacher,
Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM Journal on Control and Optimization, 39 (2000), 113-132.
doi: 10.1137/S0363012999351097. |
[4] |
J. Bonnans and H. Zidani,
Optimal control problems with partially polyhedric constraints, SIAM Journal on Control and Optimization, 37 (1999), 1726-1741.
doi: 10.1137/S0363012998333724. |
[5] |
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, 1994.
doi: 10.1007/978-1-4757-4338-8. |
[6] |
P. Das,
An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh, Numerical Algorithms, 81 (2019), 465-487.
doi: 10.1007/s11075-018-0557-4. |
[7] |
R. Dur$\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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
W. Hackbusch, Multigrid Methods and Applications, Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-02427-0. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
W. Hackbusch, Multigrid Methods and Applications, Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-02427-0. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |






4 | 8 | 16 | 32 | 64 | |
4.1470E-02 | 2.7610E-02 | 1.7622E-02 | 1.0661E-02 | 5.6118E-03 | |
order | / | 0.5869 | 0.6478 | 0.7250 | 0.9258 |
5.0583E-02 | 3.5215E-02 | 2.4487E-02 | 1.6591E-02 | 9.9351E-03 | |
order | / | 0.5225 | 0.5242 | 0.5616 | 0.7398 |
4.8403E-02 | 3.4837E-02 | 2.4420E-02 | 1.6569E-02 | 9.9076E-03 | |
order | / | 0.4745 | 0.5126 | 0.5596 | 0.7419 |
1.7466E-01 | 9.7410E-02 | 5.7862E-02 | 3.5868E-02 | 2.1591E-02 | |
order | / | 0.8424 | 0.7515 | 0.6899 | 0.7323 |
1.7207E-01 | 9.6955E-02 | 5.7777E-02 | 3.5844E-02 | 2.1562E-02 | |
order | / | 0.8276 | 0.7468 | 0.6888 | 0.7332 |
4 | 8 | 16 | 32 | 64 | |
4.1470E-02 | 2.7610E-02 | 1.7622E-02 | 1.0661E-02 | 5.6118E-03 | |
order | / | 0.5869 | 0.6478 | 0.7250 | 0.9258 |
5.0583E-02 | 3.5215E-02 | 2.4487E-02 | 1.6591E-02 | 9.9351E-03 | |
order | / | 0.5225 | 0.5242 | 0.5616 | 0.7398 |
4.8403E-02 | 3.4837E-02 | 2.4420E-02 | 1.6569E-02 | 9.9076E-03 | |
order | / | 0.4745 | 0.5126 | 0.5596 | 0.7419 |
1.7466E-01 | 9.7410E-02 | 5.7862E-02 | 3.5868E-02 | 2.1591E-02 | |
order | / | 0.8424 | 0.7515 | 0.6899 | 0.7323 |
1.7207E-01 | 9.6955E-02 | 5.7777E-02 | 3.5844E-02 | 2.1562E-02 | |
order | / | 0.8276 | 0.7468 | 0.6888 | 0.7332 |
4 | 8 | 16 | 32 | 64 | |
1.1762E-02 | 3.0109E-03 | 7.5308E-04 | 1.8747E-04 | 4.6824E-05 | |
order | / | 1.9659 | 1.9993 | 2.0062 | 2.0013 |
8.5488E-03 | 2.9582E-03 | 8.8988E-04 | 2.3807E-04 | 6.2920E-05 | |
order | / | 1.5310 | 1.7330 | 1.9022 | 1.9198 |
6.3618E-03 | 2.5354E-03 | 7.8790E-04 | 2.1187E-04 | 5.8477E-05 | |
order | / | 1.3272 | 1.6861 | 1.8949 | 1.8572 |
1.2562E-02 | 4.3649E-03 | 1.2406E-03 | 3.0455E-04 | 7.6278E-05 | |
order | / | 1.5250 | 1.8149 | 2.0263 | 1.9974 |
1.0955E-02 | 4.0558E-03 | 1.1598E-03 | 2.8267E-04 | 7.3036E-05 | |
order | / | 1.4335 | 1.8061 | 2.0367 | 1.9524 |
4 | 8 | 16 | 32 | 64 | |
1.1762E-02 | 3.0109E-03 | 7.5308E-04 | 1.8747E-04 | 4.6824E-05 | |
order | / | 1.9659 | 1.9993 | 2.0062 | 2.0013 |
8.5488E-03 | 2.9582E-03 | 8.8988E-04 | 2.3807E-04 | 6.2920E-05 | |
order | / | 1.5310 | 1.7330 | 1.9022 | 1.9198 |
6.3618E-03 | 2.5354E-03 | 7.8790E-04 | 2.1187E-04 | 5.8477E-05 | |
order | / | 1.3272 | 1.6861 | 1.8949 | 1.8572 |
1.2562E-02 | 4.3649E-03 | 1.2406E-03 | 3.0455E-04 | 7.6278E-05 | |
order | / | 1.5250 | 1.8149 | 2.0263 | 1.9974 |
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] |
Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327 |
[2] |
Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020052 |
[3] |
Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020286 |
[4] |
Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571 |
[5] |
Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020046 |
[6] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
[7] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 |
[8] |
Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399 |
[9] |
Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 |
[10] |
Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032 |
[11] |
Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 |
[12] |
Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020053 |
[13] |
Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034 |
[14] |
Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020110 |
[15] |
Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 |
[16] |
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. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 |
[17] |
Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477 |
[18] |
Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 |
[19] |
Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439 |
[20] |
Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354 |
2019 Impact Factor: 1.27
Tools
Article outline
Figures and Tables
[Back to Top]