Numerical methods for the Stokes and Navier–Stokes equations driven by threshold slip boundary conditions

Djoko, J.K.; Koko, Jonas

doi:10.1016/j.cma.2016.03.026

Cited by 40 publications

(29 citation statements)

References 33 publications

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“…In Djoko and Koko, an alternating direction method of multiplier is proposed for solving . Setting ϕ : = u τ , we consider the constrained optimization problem: Find ( u , ϕ ) such that

G false(bold-italicu false) + j_{1} false(bold-italicϕ, bold-italicϕ false) \leq G false(bold-italicv false) + j_{1} false(bold-italicφ, bold-italicφ false) \forall false(bold-italicv, bold-italicφ false)

bold-italicϕ - u_{τ} = 0 on S .

…”

Section: Alternating Direction Methods Of Multipliermentioning

confidence: 99%

“…With and , we associate the augmented Lagrangian functional

ℒ_{γ_{2}} false(bold-italicv, bold-italicϕ, normalλ false) = G false(bold-italicv false) + j_{1} false(bold-italicϕ, bold-italicϕ false) + \int_{S} false(bold-italicϕ - v_{τ} false) \cdot normalλ d σ + \frac{γ_{2}}{2} \int_{S} false| bold-italicϕ - v_{τ} |^{2} d σ,

where γ 2 > 0 is the inverse of the penalty parameter. Applying an ADMM method to we get Algorithm (see Djoko and Koko for details). We iterate until the relative error on ( u n , ϕ n ) becomes “sufficiently” small.…”

Section: Alternating Direction Methods Of Multipliermentioning

confidence: 99%

“…The “optimal” penalty parameter is γ ∗ ≈7. After some sampling procedures, we found the following “optimal” values: • γ ≈1 and γ 2 ≈30 (see), for ALG1; • γ ≈1 for ALG2; • γ ≈7 for ALG3. …”

Section: Numerical Experiments and Conclusionmentioning

confidence: 99%

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Power law Stokes equations with threshold slip boundary conditions: Numerical methods and implementation

Djoko

Koko

Kučera

2019

Math Methods in App Sciences

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For the power law Stokes equations driven by nonlinear slip boundary conditions of friction type, we propose three iterative schemes based on augmented Lagrangian approach and interior point method to solve the finite element approximation associated to the continuous problem. We formulate the variational problem which in this case is a variational inequality and construct the weak solution of the continuous problem. Next, we formulate two alternating direction methods based on augmented Lagrangian formalism in order to separate the velocity from the symmetric part the velocity gradient and tangential part of the velocity. Thirdly, we present some salient points of a path‐following variant of the interior point method associated to the finite element approximation of the problem. Some numerical experiments are performed to confirm the validity of the schemes and allow us to compare them.

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“…In Djoko and Koko, an alternating direction method of multiplier is proposed for solving . Setting ϕ : = u τ , we consider the constrained optimization problem: Find ( u , ϕ ) such that

G false(bold-italicu false) + j_{1} false(bold-italicϕ, bold-italicϕ false) \leq G false(bold-italicv false) + j_{1} false(bold-italicφ, bold-italicφ false) \forall false(bold-italicv, bold-italicφ false)

bold-italicϕ - u_{τ} = 0 on S .

…”

Section: Alternating Direction Methods Of Multipliermentioning

confidence: 99%

“…With and , we associate the augmented Lagrangian functional

ℒ_{γ_{2}} false(bold-italicv, bold-italicϕ, normalλ false) = G false(bold-italicv false) + j_{1} false(bold-italicϕ, bold-italicϕ false) + \int_{S} false(bold-italicϕ - v_{τ} false) \cdot normalλ d σ + \frac{γ_{2}}{2} \int_{S} false| bold-italicϕ - v_{τ} |^{2} d σ,

Section: Alternating Direction Methods Of Multipliermentioning

confidence: 99%

See 1 more Smart Citation

Power law Stokes equations with threshold slip boundary conditions: Numerical methods and implementation

Djoko

Koko

Kučera

2019

Math Methods in App Sciences

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“…T. Kashiwabara obtained the optimal error estimate by defining the different numerical integration of the non-differential term on the boundary S corresponding to the different finite element pairs [14,15]. J. Djoko studied the direct finite element approximation for steady Stokes problem [16] and the fully discretization scheme for nonstationary Stokes problem [17]. Y. Li and R. An discussed the stabilized and penalty finite element approximation and corresponding two-level methods for the steady Navier-Stokes equations [18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Two-level variational multiscale finite element methods for Navier–Stokes type variational inequality problem

2015

Journal of Computational and Applied Mathematics

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Variational inequality problems Variational multiscale method Two-level finite element method a b s t r a c tIn this paper, we present two-level variational multiscale finite element method based on two local Gauss integrations for Navier-Stokes equations with friction boundary conditions which are of the form of Navier-Stokes type variational inequality of the second kind. We solve Navier-Stokes type variational inequality problem on the coarse mesh and solve linearized Navier-Stokes type variational inequality problem corresponding to Newton iteration on the fine mesh. The error estimates in H 1 norm for velocity and L 2 norm for pressure are derived. Meanwhile, Uzawa iteration schemes are constructed to solve the subproblems in this two-level method. Finally, the numerical results are displayed to support the theoretical analysis.

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“…There is a vast literature on finite element methods for nonlinear elliptic and parabolic problems, for instance, previous works and references therein. The combination of numerical methods and analysis for the thermistor system for the special case κ ( u )=1 have been investigated by many authors .…”

Section: Introductionmentioning

confidence: 99%

The theta‐Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

Mbehou

2017

Math Methods in App Sciences

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This paper is devoted to the analysis of a linearized theta‐Galerkin finite element method for the time‐dependent coupled systems resulting from microsensor thermistor problems. Hereby, we focus on time discretization based on θ‐time stepping scheme with θ∈false[12,1false) including the standard Crank‐Nicolson ( θ=12) and the shifted Crank‐Nicolson ( θ=12+δ, where δ is the time‐step) schemes. The semidiscrete formulation in space is presented and optimal error bounds in L2‐norm and the energy norm are established. For the fully discrete system, the optimal error estimates are derived for the standard Crank‐Nicolson, the shifted Crank‐Nicolson, and the general case where θ≠12+kδ with k=0,1 . Finally, numerical simulations that validate the theoretical findings are exhibited.

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Numerical methods for the Stokes and Navier–Stokes equations driven by threshold slip boundary conditions

Cited by 40 publications

References 33 publications

Power law Stokes equations with threshold slip boundary conditions: Numerical methods and implementation

Power law Stokes equations with threshold slip boundary conditions: Numerical methods and implementation

Two-level variational multiscale finite element methods for Navier–Stokes type variational inequality problem

The theta‐Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

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