Use of the least squares criterion in the finite element formulation

Lynn, Paul P.; Arya, Santosh K.

doi:10.1002/nme.1620060109

Cited by 64 publications

(24 citation statements)

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“…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%

“…Among the first methods for which a decomposition (or transformation) step has been combined with least-squares principles are the methods of [72], [86], and [103]. For example, in [86], the problem (1.5)-(1.6) is transformed into a first-order div-grad system (see also [49] and [95])…”

Section: A Recipe For Practicalitymentioning

confidence: 99%

“…The first example of a div-grad decomposition has already been given by (2.11). This decomposition is also probably the first one used in the context of least-squares methods, see [72], [86], and [103]. A functional setting for the system (2.11) is provided by the space…”

Section: Discretization Levelmentioning

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: A Recipe For Practicalitymentioning

confidence: 99%

Section: A Recipe For Practicalitymentioning

confidence: 99%

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Finite Element Methods of Least-Squares Type

Bochev¹,

Gunzburger²

1998

SIAM Rev.

285

237

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“…A LSFEF (merits are well documented in Reference [4]) of Equations (4) and (5) would require that kF3C and F3C, i.e., kF and F possess the continuous "rst derivatives across the inter-element boundaries. This continuity requirement on k and can be relaxed or lowered by using the auxiliary variable approach presented by Lynn and Arya [3]. Winterscheidt and Surana [4] have presented the details of p-version LSFEF procedure for a system of non-linear partial di!erential equations.…”

Section: Least-squares Finite Element Formulation (Lsfef) Of K} Modelmentioning

confidence: 98%

p-version least-squares finite element formulation of extendedk-? model of turbulence for fully developed flow

Bagheri

Surana

2000

Commun. Numer. Meth. Engng.

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SUMMARYThis paper presents a p-version least-squares "nite element formulation for an extended k} turbulence model. The dimensionless form of the describing partial di!erential equations are cast into a system of "rst-order partial di!erential equations by utilizing auxiliary variables. Primary, and auxiliary variables are interpolated over an element using equal order C equal order p-version hierarchical interpolation functions. The least-squares error functional is constructed by taking the integrated sum of squares of the errors resulting from the system of "rst-order equations for the entire discretization. The minimization of this least-squares error functional results in "nding a solution vector + , for which the partial derivative of the error functional, with respect to nodal degrees of freedom + ,, becomes zero. This is accomplished by using Newton's method with line search. Numerical examples are presented for fully developed one-dimensional turbulent channel #ow utilizing the k} model of Launder and Sharma for various Reynolds numbers to demonstrate the convergence characteristics and accuracy of the proposed formulation. Our converged numerical solutions are in very good agreement with the computed results reported by other researchers. The formulation presented here enjoys all of the bene"ts of least-squares approach for highly non-linear and non-elliptic partial di!erential equations over Galerkin methods.

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“…Least squares ÿnite elements have also been studied for some time (see e.g. References [1][2][3][4][5][6][7][8]). Current application studies are generally directed to ÿrst-order systems (see e.g.…”

Section: Introductionmentioning

confidence: 99%