Search

This paper develops a weak Galerkin (WG) finite element method of arbitrary order for the steady incompressible Magnetohydrodynamics equations. The WG scheme uses piecewise polynomials of degrees k (k ≥ 1) , k , k − 1 and k − 1 respectively for the approximations of the velocity, the magnetic field, the pressure, and the magnetic pseudo-pressure in the interior of elements, and uses piecewise polynomials of degree k for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. We give existence and uniqueness results for the discrete scheme and derive optimal a priori error estimates. We also present a convergent linearized iterative algorithm. Numerical experiments are provided to verify the obtained theoretical results. • We develop a weak Galerkin method of arbitrary order for the steady incompressible MHD equations. • The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. • We give existence and uniqueness results for the discrete scheme and derive optimal a priori error estimates. • We also present a convergent linearized iterative algorithm. • Numerical experiments confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to propose three iterative finite element methods for equations of thermally coupled incompressible magneto-hydrodynamics (MHD) on 2D/3D bounded domain. The detailed theoretical analysis and some numerical results are presented. The main results show that the Stokes iterative method has the strictest restrictions on the physical parameters, and the Newton's iterative method has the higher accuracy and the Oseen iterative method is stable unconditionally. Design/methodology/approach: Three iterative finite element methods have been designed for the thermally coupled incompressible MHD flow on 2D/3D bounded domain. The Oseen iterative scheme includes solving a linearized steady MHD and Oseen equations; unconditional stability and optimal error estimates of numerical approximations at each iterative step are established under the uniqueness condition. Stability and convergence of numerical solutions in Newton and Stokes' iterative schemes are also analyzed under some strong uniqueness conditions. Findings: This work was supported by the NSF of China (No. 11971152). Originality/value: This paper presents the best choice for solving the steady thermally coupled MHD equations with different physical parameters. [ABSTRACT FROM AUTHOR]

Abstract: In this work, a new fully discrete stabilized finite element method is studied for the two-dimensional transient Stokes equations. This method is to use the difference between a consistent mass matrix and underintegrated mass matrix as the complement for the pressure. The spatial discretization is based on the P 1–P 1 triangular element for the approximation of the velocity and pressure, the time discretization is based on the Euler semi-implicit scheme. Some error estimates for the numerical solutions of fully discrete stabilized finite element method are derived. Finally, we provide some numerical experiments, compared with other methods, we can see that this novel stabilized method has better stability and accuracy results for the unsteady Stokes problem. [Copyright &y& Elsevier]