Computer Methods in Applied Mechanics and Engineering
Finite element approximation for grad-div type systems in the plane
SIAM Journal on Numerical Analysis
Stabilized finite element methods. I: Application to the advective-diffusive model
Computer Methods in Applied Mechanics and Engineering
Analysis of least squares finite element methods for the Stokes equations
Mathematics of Computation
First-order system least squares for second-order partial differential equations: part I
SIAM Journal on Numerical Analysis - Special issue: the articles in this issue are dedicated to Seymour V. Parter
Least-squares mixed finite elements for second-order elliptic problems
SIAM Journal on Numerical Analysis
A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Finite Element Methods of Least-Squares Type
SIAM Review
Issues Related to Least-Squares Finite Element Methods for the Stokes Equations
SIAM Journal on Scientific Computing
A Priori Error Analysis of Residual-Free Bubbles for Advection-Diffusion Problems
SIAM Journal on Numerical Analysis
Numerische Mathematik
Finite element method solution of electrically driven magnetohydrodynamic flow
Journal of Computational and Applied Mathematics
Journal of Computational Physics
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In this paper we devise a stabilized least-squares finite element method using the residual-free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow. We convert the original system of second-order partial differential equations into a first-order system formulation by introducing two additional variables. Then the least-squares finite element method using C^0 linear elements enriched with the residual-free bubble functions for all unknowns is applied to obtain approximations to the first-order system. The most advantageous features of this approach are that the resulting linear system is symmetric and positive definite, and it is capable of resolving high gradients near the layer regions without refining the mesh. Thus, this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the Hartmann number. Before incoorperating the bubble functions into the global problem, we apply the Galerkin least-squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements. Therefore, we indeed introduce a two-level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual-free bubble functions. Numerical results confirming theoretical findings are presented for several examples including the Shercliff problem.