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A Weak Galerkin Finite Element Method for the Incompressible Viscous Magneto-hydrodynamic Equations

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.

Weak Galerkin Finite Element Method, Incompressible Viscous Magneto-hydrodynamic Equations, Discrete Weak Gradient, Discrete Weak Divergence, Discrete Weak Vortex, Navier-Stokes Equations

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

AMA 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

Copyright © 2022 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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