In this paper, we are studied the weak Galerkin finite element method for the incompressible viscous Magneto-hydrodynamic (MHD) equations. There have been the several numerical methods for the incompressible viscous Magneto-hydrodynamic equations: the continuous Galerkin finite element methods, the discontinuous Galerkin finite element methods, and the mixed Galerkin finite element methods et al. These numerical methods are converted to large-scale system of linear or nonlinear equations, which are solved by direct or iterative method. Besides accuracy, efficiency, and robustness of numerical methods, physical properties such as local conservation and flux continuity are very important in practical applications. And these numerical methods have following properties. (1) The continuous Galerkin finite element methods are known as lacking of ‘‘local conservation’’. (2) The discontinuous Galerkin finite element methods are locally conservative but there is no continuity. Also the discontinuous Galerkin finite element methods increase in numbers of unknowns. (3) The mixed Galerkin finite element methods approximate the two variables that satisfy the inf-sup condition. Therefore the obtained saddle point problems are difficult to solve. A weak Galerkin finite element methods are locally conservative well, continuous across element interfaces, less unknowns than discontinuous Galerkin finite element methods, and definite discrete linear systems. A weak Galerkin finite element methods are based on new concept called discrete weak gradient, discrete weak divergence and discrete weak rotation, which are expected to play an important role in numerical methods for magneto-hydrodynamic equation. This article intends to provide a general framework for managing differential, divergence, rotation operators on generalized functions. With the proposed method, solving the magneto-hydrodynamic (MHD) equation is that the classical gradient, divergence, vortex operators are replaced by the discrete weak gradient, divergence, rotation and apply the Galerkin finite element method. It can be seen that the solution of the weak Galerkin finite element method is not only continuous function but also totally discontinuous function. For the proposed method, optimal order error estimates are established in Hilbert space.
| Published in | Engineering Mathematics (Volume 6, Issue 1) |
| DOI | 10.11648/j.engmath.20220601.12 |
| Page(s) | 6-17 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2022. Published by Science Publishing Group |
Weak Galerkin Finite Element Method, Incompressible Viscous Magneto-hydrodynamic Equations, Discrete Weak Gradient, Discrete Weak Divergence, Discrete Weak Vortex, Navier-Stokes Equations
| [1] | M. Gunzburger, A. Meir, J. Peterson, On the existence, uniqueness and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics, Math. Comput. 56 (1991) 523–563. |
| [2] | W. Layton, H. Lenferink, J. Peterson, A two-level Newton, finite element algorithm for approximating electrically conducting incompressible fluid flows, Comput. Math. Appl. 28 (1994) 21–31. |
| [3] | J. Gerbeau, A stabilized finite element method for the incompressible magnetohydrodynamic equations, Numer. Math. 87 (2000) 83–111. |
| [4] | N. Salah, A. Soulaimani, W. Habashi, A finite element method for magnetohydrodynamics, Comput. Methods Appl. Mech. Eng. 190 (2001) 5867–5892. |
| [5] | D. Schötzau, Mixed finite element methods for stationary incompressible magneto-hydrodynamics, Numer. Math. 96 (2004) 771–800. |
| [6] | G. Yuksel, R. Ingram, Numerical analysis of a finite element, Crank–Nicolson discretization for MHD flows at small magnetic Reynolds numbers, Int. J. Numer. Anal. Model. 10 (2013) 74–98. |
| [7] | X. Dong, Y. He, Y. Zhang, Convergence analysis of three finite element iterative methods for the 2D/3D stationary incompressible magnetohydrodynamics, Comput. Methods Appl. Mech. Eng. 276 (2014) 287–311. |
| [8] | Y. He, Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations, IMA J. Numer. Anal. 35 (2) (2015) 767–801. |
| [9] | H. Su, X. Feng, P. Huang, Iterative methods in penalty finite element discretization for the steady MHD equations, Comput. Methods Appl. Mech. Eng. 304 (2016) 521–545. |
| [10] | K. Hul, Y. Ma, J. Xu, Stable finite element methods preserving ∇•B=0 exactly for MHD models. Numer. Math. (2017) 135: 371–396. |
| [11] | T. Zhu, H. Su, X. Feng, Some Uzawa-type finite element iterative methods for the steady incompressible magneto-hydrodynamic equations. Appl. Math. Comput 302 (2017) 34–47. |
| [12] | R. An, C. Zhou, Error analysis of a fractional-step method for magneto-hydrodynamics equations, J. Comput. Appl. Math 313 (2017) 168–184. |
| [13] | J. Wu, D. Liu, X. Feng, P. Huang, An efficient two-step algorithm for the stationary incompressible magneto-hydrodynamic equations. Appl. Math. Comput 302 (2017) 21–33. |
| [14] | Z. Sedaghatjoo, M. Dehghan, H. Hosseinzadeh, A stable boundary elements method for magneto-hydrodynamic channel flows at high Hartmann numbers. Numer. Methods Partial Differ. Eq. (2018) 34, 575–601. |
| [15] | Y. Rong, Y. Hou, A partitioned finite element scheme based on Gauge-Uzawa method for time-dependent MHD equations. Numer. Algor (2018) 78: 277–295. |
| [16] | J. Wang, X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math. 241 (2013) 103–115. |
| [17] | L. Mu, J. Wang, X. Ye, Weak Galerkin finite element methods on Polytopal Meshes, Int. J. Numer. Anal. Model. 12 (2015) 31–53. |
| [18] | J. Wang, X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp. 83 (2014) 2101–2126. |
| [19] | L. Mu, J. Wang, X. Ye, A weak Galerkin finite element method with polynomial reduction, J. Comput. Appl. Math. 285 (2015) 45–58. |
| [20] | J. Wang, X. Ye, A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math. 42 (2016) 155–174. |
| [21] | X. Liu, J. Li, Z. Chen, A Weak Galerkin finite element method for the Oseen equations, Adv. Comput. Math. 42 (2016) 1473–1490. |
| [22] | L. Mu, J. Wang, X. Ye, A new weak Galerkin finite element method for the Helmholtz equation, IMA J. Numer. Anal. 35 (2015) 1228–1255. |
| [23] | L. Mu, J. Wang, X. Ye, S. Zhao, A numerical study on the weak Galerkin method for the Helmholtz equation, Commun. Comput. Phys. 15 (2014) 1461–1479. |
| [24] | W. Chen, F. Wang, Y. Wang, Weak Galerkin method for the coupled Darcy-Stokes flow, IMA J. Numer. Anal. 36 (2016) 897–921. |
| [25] | F. Gao, L. Mu, On Error estimate for weak Galerkin finite element methods for parabolic problems, J. Comput. Math. 32 (2014) 195–204. |
| [26] | Q. Li, J. Wang, Weak Galerkin finite element methods for parabolic equations, Numer. Methods Partial Differential Equations 29 (2013) 2004–2024. |
| [27] | F. Gao, X. Wang, A modified weak Galerkin finite element method for a class of parabolic problems, J. Comput. Appl. Math. 271 (2014) 1–19. |
| [28] | S. Y. Yi, A lowest-order weak Galerkin method for linear elasticity, J. Comput. Appl. Math, 350 (2019) 286–298. |
| [29] | R. Li, Y. Gao, J. Li, Z. Chen, A weak Galerkin finite element method for a coupled Stokes–Darcy problem on general meshes, Journal of Computational and Applied Mathematics 334 (2018) 111–127. |
| [30] | Y. Liu, J. Wang, Simplified weak Galerkin and new finite difference schemes for the Stokes equation, Journal of Computational and Applied Mathematics 361 (2019) 176–206. |
| [31] | L. Mu, Weak Galerkin based a posteriori error estimates for second order elliptic interface problems on polygonal meshes, Journal of Computational and Applied Mathematics 361 (2019) 413–425. |
| [32] | G. Chen, M. Feng, A robust WG finite element method for convection–diffusion–reaction equations, J. Comput. Appli. Math 315 (2017) 107–125. |
| [33] | F. Gao, J. Cui, G. Zhao, Weak Galerkin finite element methods for Sobolev equation, J. Comput. Appli. Math 317 (2017) 188–202. |
| [34] | R. Temam, Navier-Stokes equations; Theory and Numerical Analysis, North-Holland Publishing Company-Amsterdam, New York, Oxford, 1979. |
APA Style
Su Sung Om, Kwang Jin Kim, Thae Gun Oh, Nam Chol Yu. (2022). A Weak Galerkin Finite Element Method for the Incompressible Viscous Magneto-hydrodynamic Equations. Engineering Mathematics, 6(1), 6-17. https://doi.org/10.11648/j.engmath.20220601.12
ACS Style
Su Sung Om; Kwang Jin Kim; Thae Gun Oh; Nam Chol Yu. A Weak Galerkin Finite Element Method for the Incompressible Viscous Magneto-hydrodynamic Equations. Eng. Math. 2022, 6(1), 6-17. doi: 10.11648/j.engmath.20220601.12
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Download@article{10.11648/j.engmath.20220601.12,
author = {Su Sung Om and Kwang Jin Kim and Thae Gun Oh and Nam Chol Yu},
title = {A Weak Galerkin Finite Element Method for the Incompressible Viscous Magneto-hydrodynamic Equations},
journal = {Engineering Mathematics},
volume = {6},
number = {1},
pages = {6-17},
doi = {10.11648/j.engmath.20220601.12},
url = {https://doi.org/10.11648/j.engmath.20220601.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20220601.12},
abstract = {In this paper, we are studied the weak Galerkin finite element method for the incompressible viscous Magneto-hydrodynamic (MHD) equations. There have been the several numerical methods for the incompressible viscous Magneto-hydrodynamic equations: the continuous Galerkin finite element methods, the discontinuous Galerkin finite element methods, and the mixed Galerkin finite element methods et al. These numerical methods are converted to large-scale system of linear or nonlinear equations, which are solved by direct or iterative method. Besides accuracy, efficiency, and robustness of numerical methods, physical properties such as local conservation and flux continuity are very important in practical applications. And these numerical methods have following properties. (1) The continuous Galerkin finite element methods are known as lacking of ‘‘local conservation’’. (2) The discontinuous Galerkin finite element methods are locally conservative but there is no continuity. Also the discontinuous Galerkin finite element methods increase in numbers of unknowns. (3) The mixed Galerkin finite element methods approximate the two variables that satisfy the inf-sup condition. Therefore the obtained saddle point problems are difficult to solve. A weak Galerkin finite element methods are locally conservative well, continuous across element interfaces, less unknowns than discontinuous Galerkin finite element methods, and definite discrete linear systems. A weak Galerkin finite element methods are based on new concept called discrete weak gradient, discrete weak divergence and discrete weak rotation, which are expected to play an important role in numerical methods for magneto-hydrodynamic equation. This article intends to provide a general framework for managing differential, divergence, rotation operators on generalized functions. With the proposed method, solving the magneto-hydrodynamic (MHD) equation is that the classical gradient, divergence, vortex operators are replaced by the discrete weak gradient, divergence, rotation and apply the Galerkin finite element method. It can be seen that the solution of the weak Galerkin finite element method is not only continuous function but also totally discontinuous function. For the proposed method, optimal order error estimates are established in Hilbert space.},
year = {2022}
}
TY - JOUR T1 - A Weak Galerkin Finite Element Method for the Incompressible Viscous Magneto-hydrodynamic Equations AU - Su Sung Om AU - Kwang Jin Kim AU - Thae Gun Oh AU - Nam Chol Yu Y1 - 2022/10/18 PY - 2022 N1 - https://doi.org/10.11648/j.engmath.20220601.12 DO - 10.11648/j.engmath.20220601.12 T2 - Engineering Mathematics JF - Engineering Mathematics JO - Engineering Mathematics SP - 6 EP - 17 PB - Science Publishing Group SN - 2640-088X UR - https://doi.org/10.11648/j.engmath.20220601.12 AB - In this paper, we are studied the weak Galerkin finite element method for the incompressible viscous Magneto-hydrodynamic (MHD) equations. There have been the several numerical methods for the incompressible viscous Magneto-hydrodynamic equations: the continuous Galerkin finite element methods, the discontinuous Galerkin finite element methods, and the mixed Galerkin finite element methods et al. These numerical methods are converted to large-scale system of linear or nonlinear equations, which are solved by direct or iterative method. Besides accuracy, efficiency, and robustness of numerical methods, physical properties such as local conservation and flux continuity are very important in practical applications. And these numerical methods have following properties. (1) The continuous Galerkin finite element methods are known as lacking of ‘‘local conservation’’. (2) The discontinuous Galerkin finite element methods are locally conservative but there is no continuity. Also the discontinuous Galerkin finite element methods increase in numbers of unknowns. (3) The mixed Galerkin finite element methods approximate the two variables that satisfy the inf-sup condition. Therefore the obtained saddle point problems are difficult to solve. A weak Galerkin finite element methods are locally conservative well, continuous across element interfaces, less unknowns than discontinuous Galerkin finite element methods, and definite discrete linear systems. A weak Galerkin finite element methods are based on new concept called discrete weak gradient, discrete weak divergence and discrete weak rotation, which are expected to play an important role in numerical methods for magneto-hydrodynamic equation. This article intends to provide a general framework for managing differential, divergence, rotation operators on generalized functions. With the proposed method, solving the magneto-hydrodynamic (MHD) equation is that the classical gradient, divergence, vortex operators are replaced by the discrete weak gradient, divergence, rotation and apply the Galerkin finite element method. It can be seen that the solution of the weak Galerkin finite element method is not only continuous function but also totally discontinuous function. For the proposed method, optimal order error estimates are established in Hilbert space. VL - 6 IS - 1 ER -
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