Temporal, spatial and thermal features of 3-D Rayleigh-Bénard convection by a least-squares finite element method

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

doi:10.1016/s0045-7825(96)01053-5

Cited by 23 publications

(20 citation statements)

References 25 publications

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“…[11], [12], [15], [18], [87], [91]- [94], [96], [99], [112], and [113]. Exceptions include [13] and [14] which consider velocity gradient methods, and [92] where a stress-based method is discussed.…”

Section: H −1 Least-squares Methodsmentioning

confidence: 99%

“…The limited space did not allow us to consider many other important areas, such as hyperbolic problems, time dependent problems, and time-space leastsquares. For further details on such applications, we refer interested readers to [4], [9], [39], [40], [41], [62], [63], [88], [89], and [110]- [113], among others.…”

Section: Restricted Least-squares Methodsmentioning

confidence: 99%

“…The decomposition step consists of transforming the problem into a first-order system. For many problems, decomposition can be accomplished through the introduction of physically meaningful variables such as vorticity, stresses, or fluxes, and has been often exploited in both least-squares and Galerkin methods; see, e.g., [11]- [19], [33]- [38], [42], [43]- [45], [47]- [57], [59], [61], [62], [64], [66], [67], [72], [73], [86], [87], [90]- [96], [99], [103], [104], [107], [108], [111]- [113], and [117].…”

Section: A Recipe For Practicalitymentioning

confidence: 99%

“…The velocity-vorticity-pressure formulation of the Stokes problem. Despite the fact that it is not a fully H 1 -coercive system, the velocity-vorticitypressure formulation of the Stokes equations is, by a wide margin, the most widely used (and studied) formulation in the context of least-squares finite element method for fluid flows; see, e.g., [11], [12], [15]- [18], [34], [50], [51], [54], [57], [59], [87], [90]- [94], [96], [99], [112], and [113], among others.…”

Section: Compressible Flows and Convection-diffusion Problemsmentioning

confidence: 99%

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Finite Element Methods of Least-Squares Type

Bochev¹,

Gunzburger²

1998

SIAM Rev.

285

237

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Abstract. We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods, that are not present in standard Galerkin discretizations. Most notably, the use of least-squares principles leads to symmetric and positive definite algebraic problems and allows one to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier-Stokes equations. As a result, application of least-squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.

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Section: H −1 Least-squares Methodsmentioning

confidence: 99%

Section: Restricted Least-squares Methodsmentioning

confidence: 99%

Section: A Recipe For Practicalitymentioning

confidence: 99%

Section: Compressible Flows and Convection-diffusion Problemsmentioning

confidence: 99%

See 2 more Smart Citations

Finite Element Methods of Least-Squares Type

Bochev¹,

Gunzburger²

1998

SIAM Rev.

285

237

View full text Add to dashboard Cite

show abstract

“…After these coefficients are determined, the Galerkin series (22) satisfy all the linear homogeneous boundary conditions analytically (for details see [4]). Note also that the projection of the pressure gradient on the divergent-free basis (23), (24) is analytically zero [4], which means that the Galerkin procedure eliminates the perturbation of pressure from (14). Therefore, we do not need an approximation of the pressure.…”

Section: Methodsmentioning

confidence: 99%

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Gelfgat

1999

Journal of Computational Physics

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The article describes a complete numerical solution of a recently formulated benchmark problem devoted to the parametric study of Rayleigh-Bénard instability in rectangular two-and three-dimensional boxes. The solution is carried out by the spectral Galerkin method with globally defined, three-dimensional, divergent-free basis functions, which satisfy all boundary conditions. The general description of these three-dimensional basis functions, which can be used for a rather wide spectrum of problems, is presented. The results of the parametric calculations are presented as neutral curves showing the dependence of the critical Rayleigh number on the aspect ratio of the cavity. The neutral curves consist of several continuous branches, which belong to different modes of the most dangerous perturbation. The patterns of different perturbations are also reported. The results obtained lead to some new conclusions about the patterns of the most dangerous perturbations and about the similarities between two-and three-dimensional models. Some extensions of the considered benchmark problem are discussed.

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