The Least-Squares Finite Element Method

Jiang, Bo‐Nan

doi:10.1007/978-3-662-03740-9

Cited by 293 publications

(232 citation statements)

References 125 publications

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“…Figure 4 shows the velocity in the x-direction along the centerline of the cavity along the height of the cavity. Notice that the results correspond well with those reported in [5]. Using the NewtonSchur solution approach we were able to obtain steady state solutions up to Re = 865.…”

Section: Numerical Resultssupporting

confidence: 86%

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A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

Turner

Nakshatrala

Hjelmstad

2009

Numerical Methods in Fluids

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Abstract. In the following paper, we present a consistent Newton-Schur solution approach for variational multiscale formulations of the time-dependent Navier-Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three-dimensional problems, and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non-symmetric matrices.In addition to the quadratic convergence characteristics of a Newton-Raphson based scheme, the Newton-Schur approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur's complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two-level approach to stabilizing the incompressible Navier-Stokes equations based on a coarse and finescale subproblem. We then derive the Schur's complement form of the consistent tangent matrix.We demonstrate the performance of the method for a number of three-dimensional problems for Reynolds number up to 1000 including steady and time-dependent flows.

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Section: Numerical Resultssupporting

confidence: 86%

“…These instabilities represent the transition into turbulent flow [28]. The failure to converge above Re = 800 for the Newton-Schur solution strategy could be due to the presence of TaylorGörtler vortices as reported in [5].…”

Section: Numerical Resultsmentioning

confidence: 86%

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

Turner

Nakshatrala

Hjelmstad

2009

Numerical Methods in Fluids

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“…Consider the backward-facing step domain shown in figure 3.7. An analysis of the least-squares method for Navier-Stokes equations on such a domain can be found in [11]. Our prototype domain has the ratio 1 × 1 × 10 and the step has a height 0.5 and a length 2, that is, a 1 × 0.5 × 2 notch of the bottom corner is removed.…”

Section: 4mentioning

confidence: 99%

“…Introduction. The finite element method based on first-order system leastsquares (FOSLS) formulation has been used in a wide variety of applications [3], [4], [5], [6], [11]. We are interested, here, in least-squares finite element methods for Stokes and Navier-Stokes equations.…”

mentioning

confidence: 99%

Enhanced Mass Conservation in Least-Squares Methods for Navier–Stokes Equations

Heys¹,

Lee²,

Manteuffel³

et al. 2009

SIAM J. Sci. Comput.

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Abstract. There are many applications of the least-squares finite element method for the numerical solution of partial differential equations because of a number of benefits that the leastsquares method has. However, one of most well-known drawbacks of the least-squares finite element method is the lack of exact discrete mass conservation, in some contexts, due to the fact that leastsquares method minimizes the continuity equation in L 2 norm. In this paper, we explore the reason of the mass loss and provide new approaches to retain the mass even in severely under-resolved grid. . On the other hand, least-squares methods for Stokes and Navier-Stokes equations lack an exact sense of discrete mass conservation. More precisely, least-squares methods do not naturally constrain the velocities to exactly satisfy any form of the mass conservation equations on discrete volumes as Galerkin and finite volume methods typically do. Instead, these methods merely strike a balance between the L 2 errors in each equation, with no special attention to the one that expresses mass conservation. Conservation error generally tends to vanish as the resolution or approximation order increases, but some applications require a level of discrete conservation that substantially exceeds the overall accuracy obtained by the discretization. In particular, for problems in which the domain is long and thin and only velocity boundary conditions are prescribed on all but a relatively small portion of the boundary, the use of low-order finite element spaces with a relatively coarse grid without further mitigation can result in severe mass loss. This deficiency can be overcome in two-dimensional problems by using higher-order finite element spaces, as the results in [15] show. However, in three dimensions, further measures may be necessary, which is the focus of this paper.In [9], mass conservation was improved by introducing newly defined variables and corresponding boundary conditions. Here, we instead suggest more fundamental ways of achieving improved mass conservation. We first identify a cause of the loss of mass in the least-squares finite element method in three-dimensional problems. In section 3, we show that boundary conditions near edges parallel to the flow contribute to mass-loss, even when high-order finite elements are employed. In section 3.1, a

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“…Method which is more commonly used for the fluid transport equation is the finite element method. Among some types of finite element methods, Least Squares method is more commonly used especially in the field of fluid mechanics and electromagnetics [2]. The Least Square finite element method is based on minimizing second rank residuals.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation of Heat Transfer Diffusion-Convection in Two-Dimensional Space

Purnami¹,

Made²,

Suar³

2018

IJCA

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The diffusion convection equation is a mathematical model of transport event that occurs in the fluid. The equation can be solved numerically by using finite element method and in this research use Least Square finite element method. This method is based on minimizing second cube residue and on its application, it will produce a symmetric system and definite positive linear equations. This research will present a numerical simulation on the diffusion convection equation in two dimensional space use linear interpolation.

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The Least-Squares Finite Element Method

Cited by 293 publications

References 125 publications

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

Enhanced Mass Conservation in Least-Squares Methods for Navier–Stokes Equations

Numerical Simulation of Heat Transfer Diffusion-Convection in Two-Dimensional Space

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