Unconditional stability and convergence of fully discrete schemes for $2D$ viscous fluids models with mass diffusion

Guillén-González, Francisco; Gutiérrez-Santacreu, Juan Vicente

doi:10.1090/s0025-5718-08-02099-1

Cited by 19 publications

(35 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since a maximum principle cannot be verified in general by the discrete concentration, we introduce a truncation operator on the L 2 projection onto P 0 , in order to guarantee a L ∞ bound for some terms in the discrete concentration equation. A similar idea of truncation, but without the L 2 projection onto P 0 , has been used in [8] for a 2D Navier-Stokes model with mass diffusion.…”

Section: Main Results Of the Papermentioning

confidence: 99%

See 1 more Smart Citation

Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model

Guillén-González¹,

Gutiérrez-Santacreu²

2009

ESAIM: M2AN

Self Cite

View full text Add to dashboard Cite

Abstract. We analyze two numerical schemes of Euler type in time and C 0 finite-element type with P1-approximation in space for solving a phase-field model of a binary alloy with thermal properties. This model is written as a highly non-linear parabolic system with three unknowns: phase-field, solute concentration and temperature, where the diffusion for the temperature and solute concentration may degenerate. The first scheme is nonlinear, unconditionally stable and convergent. The other scheme is linear but conditionally stable and convergent. A maximum principle is avoided in both schemes, using a truncation operator on the L 2 projection onto the P0 finite element for the discrete concentration. In addition, for the model when the heat conductivity and solute diffusion coefficients are constants, optimal error estimates for both schemes are shown based on stability estimates.

show abstract

Section: Main Results Of the Papermentioning

confidence: 99%

“…This compactness is not clear if we truncate c h,k by nodes, as was made in [8] for a nondegenerate mass diffusion Navier-Stokes model.…”

Section: Proposition 42 If the Function G 2 (H) Given In Section 2 mentioning

confidence: 99%

Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model

Guillén-González¹,

Gutiérrez-Santacreu²

2009

ESAIM: M2AN

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recently, the existence and regularity of strong solutions have been proved in [10] by means of an iterative method (jointly with some error estimates). A time-space numerical scheme has been recently developed by using C 0 -finite elements for density and velocity in [11] for model (1.1) in 2D domains, which is unconditionally stable and convergent towards the (unique) weak solution of the continuous problem. This scheme is of the backward Euler type, where in each time step the computation of the density and the velocity pressure are decoupled, by means of linear problems.…”

Section: Known Resultsmentioning

confidence: 99%

Conditional Stability and Convergence of a Fully Discrete Scheme for Three-Dimensional Navier–Stokes Equations with Mass Diffusion

Guillén-González¹,

Gutiérrez-Santacreu²

2008

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract. We construct a fully discrete numerical scheme for three-dimensional incompressible fluids with mass diffusion (in density-velocity-pressure formulation), also called the KazhikhovSmagulov model. We will prove conditional stability and convergence, by using at most C 0 -finite elements, although the density of the limit problem will have H 2 -regularity. The key idea of our argument is first to obtain pointwise estimates for the discrete density by imposing the constraint lim (h,k)→0 h/k = 0 on the time and space parameters (k, h). Afterwards, under the same constraint on the parameters, strong estimates for the discrete density in l ∞ (H 1 ) and for the discrete Laplacian of the density in l 2 (L 2 ) are obtained. From here, the compactness and convergence of the scheme can be concluded with similar arguments as we used in [Math. Comp., to appear], where a different scheme is studied for two-dimensional domains which is unconditionally stable and convergent. Moreover, we study the asymptotic behavior of the numerical scheme as the diffusion parameter λ goes to zero, obtaining convergence as (k, h, λ) → 0 towards a weak solution of the density-dependent Navier-Stokes system provided that the constraint lim (λ,h,k)→0 h/(λ 2 k) = 0 on (h, k, λ) is satisfied.

show abstract

“…is proved in [28], under the regularity (32) with r > d for A −1 the Stokes operator (31). Thus, applying Sobolev's inequality, H 2 (Ω) ֒→ W 1,r (Ω), with r ≤ 6, gives…”

Section: Using W = −∆D + F ε (D) As An Auxiliary Variablementioning

confidence: 88%

“…Obtaining a compactness result for the discrete velocity turns out to be harder than for the GinzburgLandau problem (11) for a fixed ε, even for problems with a similar structure such as the density-dependent Navier-Stokes equations ( [50], [31] and [32]). Now we cannot prove the time fractional estimate…”

Section: Using W = −∆D + F ε (D) As An Auxiliary Variablementioning

confidence: 99%

An Overview on Numerical Analyses of Nematic Liquid Crystal Flows

Badia

Guillén-González

Gutiérrez-Santacreu

2011

Arch Computat Methods Eng

View full text Add to dashboard Cite

The purpose of this work is to provide an overview of the most recent numerical developments in the field of nematic liquid crystals. The Ericksen-Leslie equations govern the motion of a nematic liquid crystal. This system, in its simplest form, consists of the NavierStokes equations coupled with an extra anisotropic stress tensor, which represents the effect of the nematic liquid crystal on the fluid, and a convective harmonic map equation. The sphere constraint must be enforced almost everywhere in order to obtain an energy estimate.Since an almost everywhere satisfaction of this restriction is not appropriate at a numerical level, two alternative approaches have been introduced: a penalty method and a saddle-point method. These approaches are suitable for their numerical approximation by finite elements, since a discrete version of the restriction is enough to prove the desired energy estimate.The Ginzburg-Landau penalty function is usually used to enforce the sphere constraint.Finite element methods of mixed type will play an important role when designing numerical approximations for the penalty method in order to preserve the intrinsic energy estimate.The inf-sup condition that makes the saddle-point method well-posed is not clear yet.The only inf-sup condition for the Lagrange multiplier is obtained in the dual space of H 1 (Ω).But such an inf-sup condition requires more regularity for the director vector than the one provided by the energy estimate. Herein, we will present an alternative inf-sup condition whose proof for its discrete counterpart with finite elements is still open.2000 Mathematics Subject Classification. 35Q35, 65M12, 65M60.

show abstract

Unconditional stability and convergence of fully discrete schemes for $2D$ viscous fluids models with mass diffusion

Cited by 19 publications

References 14 publications

Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model

Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model

Conditional Stability and Convergence of a Fully Discrete Scheme for Three-Dimensional Navier–Stokes Equations with Mass Diffusion

An Overview on Numerical Analyses of Nematic Liquid Crystal Flows

Contact Info

Product

Resources

About