
-
Previous Article
Observer-based control for a class of hybrid linear and nonlinear systems
- DCDS-S Home
- This Issue
-
Next Article
Optimal synchronization control of multiple euler-lagrange systems via event-triggered reinforcement learning
Parallelization of a finite volumes discretization for anisotropic diffusion problems using an improved Schur complement technique
1. | Abdelmalek Essaadi University, Faculty of Sciences and Techniques of Tangier, Laboratory of Mathematics and Applications, Department of Mathematics, BP. 416, Tangier 90000, Morocco |
2. | Moulay Ismail University of Meknes Faculty of Sciences and Techniques of Errachidia Department of Mathematics MSISI Laboratory, AM2CSI Group BP. 509, Boutalamine Errachidia 57000, Morocco |
3. | Moulay Ismail University of Meknes, Faculty of Sciences of Meknes, Department of Mathematics and Informatics, BP. 11201 Zitoune Meknes, Morocco |
We present in this paper a new algorithm combining a finite volume method with an improved Schur complement technique to solve $ 2D $ anisotropic diffusion problems on general meshes. After having proved the convergence of the finite volume method, we have given a description of the proposed algorithm in the case of two nonoverlapping subdomains. Several numerical tests are achieved which illustrate the theoretical results of convergence of the finite volume method and show the advantages of the proposed algorithm.
References:
[1] |
L. Agelas and R. Masson,
Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes, C. R. Acad. Sci. Paris. Ser. I, 346 (2008), 1007-1012.
doi: 10.1016/j.crma.2008.07.015. |
[2] |
B. Andreianov, F. Boyer and F. Hubert,
Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general $2D$ meshes, Num. Meth. PDE., 23 (2007), 145-195.
doi: 10.1002/num.20170. |
[3] |
H. Belhadj, M. Fihri, S. Khallouq and N. Nagid,
Optimal number of schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 21-34.
doi: 10.3934/dcdss.2018002. |
[4] |
P. E. Bjørstad and O. B. Widlund,
Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal., 23 (1986), 1097-1120.
doi: 10.1137/0723075. |
[5] |
F. Boyer, F. Hubert and S. Krell,
Non-overlapping Schwarz algorithm for solving two-dimensional m-DDFV schemes, IMA J. Numer. Anal., 30 (2010), 1062-1100.
doi: 10.1093/imanum/drp001. |
[6] |
S. C. Brenner,
The condition number of the Schur complement in domain decomposition, Numer. Math., 83 (1999), 187-203.
doi: 10.1007/s002110050446. |
[7] |
T. F. Chan, E. Weinan and J. Sun,
Domain decomposition interface preconditioners for fourth-order elliptic problems, Appl. Numer. Math., 8 (1991), 317-331.
doi: 10.1016/0168-9274(91)90072-8. |
[8] |
Y. Coudière, J.-P. Vila and P. Villedieu,
Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem, ESAIM:M2AN, 33 (1999), 493-516.
doi: 10.1051/m2an:1999149. |
[9] |
K. Domelevo and P. Omnès,
A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, ESAIM:M2AN, 39 (2005), 1203-1249.
doi: 10.1051/m2an:2005047. |
[10] |
R. Eymard, T. Gallouët and R. Herbin,
Convergence of finite volume schemes for semilinear convection diffusion equations, Numer. Math., 82 (1999), 91-116.
doi: 10.1007/s002110050412. |
[11] |
R. Eymard, T. Gallouët and R. Herbin,
Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: A scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), 1009-1043.
doi: 10.1093/imanum/drn084. |
[12] |
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis (eds. P.G. Ciarlet and J.L. Lions), Elsevier, 7 (2000), 713–1020. |
[13] |
R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in ISTE. Finite Volumes for Complex Applications V (eds. R. Eymard and J.-M. Hérard), Wiley, 5 (2008), 659–692. |
[14] |
F. Hermeline,
A finite volume method for the approximation of diffusion operators on distorted meshes, J. Comput. Phys., 160 (2000), 481-499.
doi: 10.1006/jcph.2000.6466. |
[15] |
L. Mansfield,
On the conjugate gradient solution of the Schur complement system obtained from domain decomposition, SIAM J. Numer. Anal., 27 (1990), 1612-1620.
doi: 10.1137/0727094. |
[16] |
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999.
![]() |
show all references
References:
[1] |
L. Agelas and R. Masson,
Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes, C. R. Acad. Sci. Paris. Ser. I, 346 (2008), 1007-1012.
doi: 10.1016/j.crma.2008.07.015. |
[2] |
B. Andreianov, F. Boyer and F. Hubert,
Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general $2D$ meshes, Num. Meth. PDE., 23 (2007), 145-195.
doi: 10.1002/num.20170. |
[3] |
H. Belhadj, M. Fihri, S. Khallouq and N. Nagid,
Optimal number of schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 21-34.
doi: 10.3934/dcdss.2018002. |
[4] |
P. E. Bjørstad and O. B. Widlund,
Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal., 23 (1986), 1097-1120.
doi: 10.1137/0723075. |
[5] |
F. Boyer, F. Hubert and S. Krell,
Non-overlapping Schwarz algorithm for solving two-dimensional m-DDFV schemes, IMA J. Numer. Anal., 30 (2010), 1062-1100.
doi: 10.1093/imanum/drp001. |
[6] |
S. C. Brenner,
The condition number of the Schur complement in domain decomposition, Numer. Math., 83 (1999), 187-203.
doi: 10.1007/s002110050446. |
[7] |
T. F. Chan, E. Weinan and J. Sun,
Domain decomposition interface preconditioners for fourth-order elliptic problems, Appl. Numer. Math., 8 (1991), 317-331.
doi: 10.1016/0168-9274(91)90072-8. |
[8] |
Y. Coudière, J.-P. Vila and P. Villedieu,
Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem, ESAIM:M2AN, 33 (1999), 493-516.
doi: 10.1051/m2an:1999149. |
[9] |
K. Domelevo and P. Omnès,
A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, ESAIM:M2AN, 39 (2005), 1203-1249.
doi: 10.1051/m2an:2005047. |
[10] |
R. Eymard, T. Gallouët and R. Herbin,
Convergence of finite volume schemes for semilinear convection diffusion equations, Numer. Math., 82 (1999), 91-116.
doi: 10.1007/s002110050412. |
[11] |
R. Eymard, T. Gallouët and R. Herbin,
Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: A scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), 1009-1043.
doi: 10.1093/imanum/drn084. |
[12] |
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis (eds. P.G. Ciarlet and J.L. Lions), Elsevier, 7 (2000), 713–1020. |
[13] |
R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in ISTE. Finite Volumes for Complex Applications V (eds. R. Eymard and J.-M. Hérard), Wiley, 5 (2008), 659–692. |
[14] |
F. Hermeline,
A finite volume method for the approximation of diffusion operators on distorted meshes, J. Comput. Phys., 160 (2000), 481-499.
doi: 10.1006/jcph.2000.6466. |
[15] |
L. Mansfield,
On the conjugate gradient solution of the Schur complement system obtained from domain decomposition, SIAM J. Numer. Anal., 27 (1990), 1612-1620.
doi: 10.1137/0727094. |
[16] |
A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999.
![]() |





Mesh | Nbr of |
|||||
- | - | |||||
Average |
Mesh | Nbr of |
|||||
- | - | |||||
Average |
Mesh | Nbr of |
|||||
- | - | |||||
Average |
Mesh | Nbr of |
|||||
- | - | |||||
Average |
Mesh | Nbr of |
|||||
Average |
Mesh | Nbr of |
|||||
Average |
Mesh | Nbr of |
|||||
Average |
Mesh | Nbr of |
|||||
Average |
Mesh | Nbr of |
Nbr of |
||||
Mesh | Nbr of |
Nbr of |
||||
Nbr of |
CPU time for DDFV (in seconds) | CPU time for DDFV-SC (in seconds) |
Nbr of |
CPU time for DDFV (in seconds) | CPU time for DDFV-SC (in seconds) |
[1] |
Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 |
[2] |
Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096 |
[3] |
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 |
[4] |
Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021018 |
[5] |
Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127 |
[6] |
Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020319 |
[7] |
Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331 |
[8] |
Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072 |
[9] |
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 |
[10] |
Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020321 |
[11] |
Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 |
[12] |
Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 |
[13] |
Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126 |
[14] |
Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073 |
[15] |
Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326 |
[16] |
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 |
[17] |
Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 |
[18] |
Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105 |
[19] |
Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 |
[20] |
Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 |
2019 Impact Factor: 1.233
Tools
Article outline
Figures and Tables
[Back to Top]