A least‐squares finite element method for time‐dependent incompressible flows with thermal convection

Tang, Li Qiang; Tsang, Tate T. H.

doi:10.1002/fld.1650170402

Cited by 63 publications

(42 citation statements)

References 31 publications

(6 reference statements)

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“…Initially some verification calculations were performed for pure hydrodynamic case taken Mnf = 0 and the results are well agreed with [16]. The purpose is to assess the stability and accuracy of the finite element algorithm.…”

Section: Numerical Results and Discussionmentioning

confidence: 69%

Mixed Finite Element Formulation for Magnetic Fluid Oil Flow in Electromagnetic Field

Tan

Yassin

2017

MATEC Web Conf.

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Abstract. Pressure depletion and high viscosity of crude oil in oil reservoir are the main challenges in oil recovery process. A potential solution is to employ electromagnetic heating coupled with magnetic fluid injection. The present work delivers a fundamental study on the interaction between magnetic fluid flow with electromagnetic field. The two-dimensional, incompressible flow is solved numerically using mixed finite element method. The velocity fields, temperature and pressure are the variables of interest, to be obtained by solving mass, momentum and energy equations coupled with Maxwell' equations. The fluid stress arises simultaneously with the external magnetic force which mobilises and increases the temperature of the oil flow. Verification is made against available data obtained from different numerical method reported in literature. The results justify feasibility of the mixed finite element formulation as an alternative for the modelling of the magnetic fluid flow.

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Section: Numerical Results and Discussionmentioning

confidence: 69%

Mixed Finite Element Formulation for Magnetic Fluid Oil Flow in Electromagnetic Field

Tan

Yassin

2017

MATEC Web Conf.

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“…However, even the improved mass conservation is still not acceptable, and it came at a increased computational cost (e.g., if the one-dimensional polynomial order is p, then computational costs scale as O(p 6 ) if exact integration is used). Related to the idea of using alternative finite element spaces is the idea of using reduced integration methods [21,26]. Often, the use of reduced integration results in a collocation solution [6,24].…”

Section: Possible Remediesmentioning

confidence: 99%

On Mass‐Conserving Least‐Squares Methods

Heys¹,

Lee²,

Manteuffel³

et al. 2006

SIAM J. Sci. Comput.

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Abstract. Least-squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations, including symmetric positive definite discrete operators and a sharp error measure. One of the potential drawbacks, especially in three dimensions, is that mass conservation is achieved only in a least-squares sense, and underresolved solutions are especially vulnerable to poor conservation. For the stationary Navier-Stokes equations, which are typically rewritten as a larger system of first-order equations, the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used. A new first-order system reformulation of the Navier-Stokes equations is presented that admits a wider range of mass-conserving boundary conditions. This new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems, a square channel, backwardfacing step, and branching tubes with two generations of bifurcations.

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“…One easily sees that (27) is equivalent to the variational problem: (28) findUh £ UA such that &(Uh, Vh) = &(Vh) Wh £ \Jh ;…”

Section: Jamentioning

confidence: 99%

“…Recently, there has been substantial interest in the use of least squares principles for the approximate solution of the Navier-Stokes equations of incompressible flow; for some examples of bona fide least squares methods, one may consult, e.g., [5,8,9,11,12,19,20,21,22,23,24,28]. The computational results provided in these papers indicate that the methods considered are effective; however, careful analyses of these methods indicate that they do not yield optimally accurate approximations.…”

Section: Introductionmentioning

confidence: 99%

Analysis of least squares finite element methods for the Stokes equations

Bochev¹,

Gunzburger²

1994

Math. Comp.

121

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Abstract. In this paper we consider the application of least squares principles to the approximate solution of the Stokes equations cast into a first-order velocity-vorticity-pressure system. Among the most attractive features of the resulting methods are that the choice of approximating spaces is not subject to the LBB condition and a single continuous piecewise polynomial space can be used for the approximation of all unknowns, that the resulting discretized problems involve only symmetric, positive definite systems of algebraic equations, that no artificial boundary conditions for the vorticity need be devised, and that accurate approximations are obtained for all variables, including the vorticity. Here we study two classes of least squares methods for the velocity-vorticitypressure equations. The first one uses norms prescribed by the a priori estimates of Agmon, Doughs, and Nirenberg and can be analyzed in a completely standard manner. However, conforming discretizations of these methods require C1 continuity of the finite element spaces, thus negating the advantages of the velocity-vorticity-pressure formulation. The second class uses weighted L2-norms of the residuals to circumvent this flaw. For properly chosen meshdependent weights, it is shown that the approximations to the solutions of the Stokes equations are of optimal order. The results of some computational experiments are also provided; these illustrate, among other things, the necessity of introducing the weights.

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A least‐squares finite element method for time‐dependent incompressible flows with thermal convection

Cited by 63 publications

References 31 publications

Mixed Finite Element Formulation for Magnetic Fluid Oil Flow in Electromagnetic Field

Mixed Finite Element Formulation for Magnetic Fluid Oil Flow in Electromagnetic Field

On Mass‐Conserving Least‐Squares Methods

Analysis of least squares finite element methods for the Stokes equations

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