On mass conservation in least-squares methods

Bolton, Patrick; Thatcher, R. W.

doi:10.1016/j.jcp.2004.08.013

Cited by 34 publications

(24 citation statements)

References 29 publications

(69 reference statements)

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“…Similar results using H −1 -norm least-squares functional for Stokes equations based on the velocity-pressure-stress formulation are presented in [8]. Regardless of their advantages, poor mass conservation is reported in least-squares based formulations, see e.g., [9][10][11]. Note, however, as indicated in [10], results can be improved by sufficiently weighting the mass conservation term.…”

Section: Introductionsupporting

confidence: 56%

“…6, the Union Jack grid with the uniform mesh spacing of 0.0179 on [−2, 2] and 0.025 on [2,5] is considered. In [10], such grid is illustrated to have special properties not necessary possessed by other configurations. The computational results are presented based on the velocity component along the axis of symmetry and corner vortex behaviors.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Regardless of their advantages, poor mass conservation is reported in least-squares based formulations, see e.g., [9][10][11]. Note, however, as indicated in [10], results can be improved by sufficiently weighting the mass conservation term.…”

Section: Introductionmentioning

confidence: 99%

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A nonlinear weighted least-squares finite element method for Stokes equations

Lee

Chen

2010

Computers & Mathematics with Applications

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a b s t r a c tThe paper concerns a nonlinear weighted least-squares finite element method for the solutions of the incompressible Stokes equations based on the application of the leastsquares minimization principle to an equivalent first order velocity-pressure-stress system. Model problem considered is the flow in a planar channel. The least-squares functional involves the L 2 -norms of the residuals of each equation multiplied by a nonlinear weighting function and mesh dependent weights. Using linear approximations for all variables, by properly adjusting the importance of the mass conservation equation and a carefully chosen nonlinear weighting function, the least-squares solutions exhibit optimal L 2 -norm error convergence in all unknowns. Numerical solutions of the flow pass through a 4 to 1 contraction channel will also be considered.

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Section: Introductionsupporting

confidence: 56%

Section: Numerical Resultsmentioning

confidence: 99%

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A nonlinear weighted least-squares finite element method for Stokes equations

Lee

Chen

2010

Computers & Mathematics with Applications

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“…However, this changed the least-squares principle to a saddle-point problem, thereby negating one of the key advantages of LSFEMs over other methods. Subsequently, mass conservation in least-squares methods for the Stokes and the Navier-Stokes equations has been studied extensively in the literature [2][3][4][5][6][7][8][9].…”

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confidence: 99%

A locally conservative, discontinuous least‐squares finite element method for the Stokes equations

Bochev

Lai

Olson

2011

Numerical Methods in Fluids

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SUMMARYConventional least-squares finite element methods (LSFEMs) for incompressible flows conserve mass only approximately. For some problems, mass loss levels are large and result in unphysical solutions. In this paper we formulate a new, locally conservative LSFEM for the Stokes equations wherein a discrete velocity field is computed that is point-wise divergence free on each element. The central idea is to allow discontinuous velocity approximations and then to define the velocity field on each element using a local stream-function. The effect of the new LSFEM approach on improved local and global mass conservation is compared with a conventional LSFEM for the Stokes equations employing standard C 0 Lagrangian elements.

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“…For example, making β larger (i.e., less error in conservation of mass) can result in faster or slower multigrid convergence, depending on the Reynolds number, Re, and the actual functional used (G is not a computationally practical functional and is rarely used). The effects of weighting the functional are demonstrated later in this paper, but, even though this can be beneficial in some situations, it should not be considered a universal solution [8].…”

mentioning

confidence: 98%

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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On mass conservation in least-squares methods

Cited by 34 publications

References 29 publications

A nonlinear weighted least-squares finite element method for Stokes equations

A nonlinear weighted least-squares finite element method for Stokes equations

A locally conservative, discontinuous least‐squares finite element method for the Stokes equations

On Mass‐Conserving Least‐Squares Methods

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