An ‘upwind’ finite element scheme for two‐dimensional convective transport equation

Heinrich, J. C.; Huyakorn, Peter S.; Zienkiewicz, O. C.; Mitchell, A. R.

doi:10.1002/nme.1620110113

Cited by 529 publications

(153 citation statements)

References 7 publications

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“…An "up-wind" finite element scheme improved from the Galerkin finite element method is applied to overcome the difficulties caused by the convection terms in some cases with relative larger value of parameter [24]. In an element, the variables are the linear combinations of node values, expressed as the follows:…”

Section: Numerical Simulationmentioning

confidence: 99%

Crystal growth in floating zone with phase change and thermo-solutal convections

Xiong

1992

Journal of Crystal Growth

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Floating zone crystal growth in microgravity environment is investigated numerically by a finite element method for semiconductor growth processing, which involves thermocapillary convection, phase change convection, thermal diffusion and solutal diffusion. The configurations of phase change interfaces and distributions of velocity, temperature and concentration fields are analyzed for typical conditions of pulling rates and segregation coefficients. The influence of phase change convection on the distribution of concentration is studied in detail. The results show that the thermocapillary convection plays an important role in mixing up the melt with dopant. The deformations of phase change interfaces by thermal convection-diffusion and pulling rods make larger variation of concentration field in comparison with the case of plane interfaces.

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Section: Numerical Simulationmentioning

confidence: 99%

Crystal growth in floating zone with phase change and thermo-solutal convections

Xiong

1992

Journal of Crystal Growth

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“…In the ··1 ast few years increasing number of numeri ca 1 schemes have appeared. Recent advances in both finite differences [Chaudhari, 1971;Todd, 1972] and finite elements [Heinrich et al, 1977;Huyakorn and Nilkuha, 1979] have developed so far as eliminating numerical difficulties encountered in the higher ranges of Peclet number, resulting from the oscillatory nature of the governing differential equation. To our knowledge, all the existing numerical techniques are concerned with only continuum applicationsc Although, these.…”

Section: Solution Approachmentioning

confidence: 99%

“…Since the formulation and use of upstream finite element method has been addressed adequately [Heinrich et al, 1977;Huyakorn and Nilkuha, 1979],-the numerical procedures for the matrix part of the domain of the solute transport problem are given briefly while those concerning fractures will be discussed in detail.…”

Section: Weighted Residual Finite Element Formulationmentioning

confidence: 99%

An upstream finite element method for solution of transient transport equation in fractured porous media

Noorishad¹,

Mehran²

1982

Water Resources Research

298

108

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A finite element method for the solution of two‐dimensional transient dispersive‐convective transport of nonconservative solute species in fractured porous media is presented. A two‐nodal point one‐dimensional transport element for fractures is developed which provides a number of advantages relative to conventional fracture representation by two‐dimensional continuum elements. To eliminate the oscillatory behavior of convective‐dominated transport which is a more likely occurrence in fracture, a very efficient one‐dimensional upstreaming method along with a two‐dimensional method is implemented. Validity of the numerical scheme is established by comparison with existing one‐ and two‐dimensional analytic solutions.

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“…To overcome this stability problem, many methods have been proposed. These include upwind methods (cf., for example, [5,11,12,4,2]), streamline diffusion methods (cf., for example, [13]) and exponentially fitted methods (cf., for example, [14,15,18,23,24]). However, no method guarantees, in general, that a numerical solution converges to the exact one uniformly in ε on an unstructured triangular partition.…”

Section: Introductionmentioning

confidence: 99%

A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

Wang

2003

Math. Comp.

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Abstract. In this paper we formulate and analyze a discretization method for a 2D linear singularly perturbed convection-diffusion problem with a singular perturbation parameter ε. The method is based on a nonconforming combination of the conventional Galerkin piecewise linear triangular finite element method and an exponentially fitted finite volume method, and on a mixture of triangular and rectangular elements. It is shown that the method is stable with respect to a semi-discrete energy norm and the approximation error in the semi-discrete energy norm is bounded by Ch ln ε ln h with C independent of the mesh parameter h, the diffusion coefficient ε and the exact solution of the problem.

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An ‘upwind’ finite element scheme for two‐dimensional convective transport equation

Cited by 529 publications

References 7 publications

Crystal growth in floating zone with phase change and thermo-solutal convections

Crystal growth in floating zone with phase change and thermo-solutal convections

An upstream finite element method for solution of transient transport equation in fractured porous media

A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

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