Penalty finite element method for Stokes problem with nonlinear slip boundary conditions

Li, Yuan; Li, Kaitai

doi:10.1016/j.amc.2008.06.035

Cited by 28 publications

(26 citation statements)

References 14 publications

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“…Following the proof of Theorem 2.1, we also have the following theorem about the solution of the regularized problem (20).…”

Section: The Regularized Problemmentioning

confidence: 99%

“…Y. Li and K.T. Li study the penalty finite element approximation to the Stokes problem [20] and study the pressure projection stabilized finite element approximation to the steady Navier-Stokes problem [21] and derive the optimal error estimates under the strong regularity assumption on the velocity field. In these papers, in order to construct the computational iteration schemes, the multiplier also is introduced.…”

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confidence: 99%

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Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

2010

Numer. Math.

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Based on the pressure projection stabilized methods, the semi-discrete finite element approximation to the time-dependent Navier-Stokes equations with nonlinear slip boundary conditions is considered in this paper. Because this class of boundary condition includes the subdifferential property, then the variational formulation is the Navier-Stokes type variational inequality problem. Using the regularization procedure, we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter ε, which means that the solution of the regularized problem converges to the solution of the Navier-Stokes type variational inequality problem as the parameter ε −→ 0. Moreover, some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data. The pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived.

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“…Following the proof of Theorem 2.1, we also have the following theorem about the solution of the regularized problem (20).…”

Section: The Regularized Problemmentioning

confidence: 99%

mentioning

confidence: 99%

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

2010

Numer. Math.

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“…The numerical investigation of these later problems have benefited a lot from the existing knowledge of general ways of treating variational inequalities and some works worth to be mentioned here include [2,7,18,30,39,40].…”

Section: Introductionmentioning

confidence: 99%

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

Djoko

Koko

2016

Computer Methods in Applied Mechanics and Engineering

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In this article, we discuss the numerical solution of the Stokes and Navier-Stokes equations completed by nonlinear slip boundary conditions of friction type in two and three dimensions. To solve the Stokes system, we first reduce the related variational inequality into a saddle point-point problem for a well chosen augmented Lagrangian. To solve this saddle point problem we suggest an alternating direction method of multiplier together with finite element approximations. The solution of the Navier Stokes system combines finite element approximations, time discretization by operator splitting and augmented Lagrangian method. Numerical experiment results for two and three dimensional flow confirm the interest of these approaches.

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“…The finite element methods presented in [15,18] are motivated by problems in plasticity, while the analysis in [17] uses the penalty approach in the Stokes equations to circumvent the incompressibility constraint. Using a different type of slip boundary condition R. Verfurth [20] has analyzed the problem by relaxing the constraint (1.4) at the expense of an additional unknown.…”

Section: Introduction We Consider Steady Flows Of Incompressible Vismentioning

confidence: 99%

“…There exist many finite element discretizations for solving variational inequalities [13,14,19], steady Stokes and Navier-Stokes problems [10,13] and mixed variational inequalities [9,15,17,18]. The finite element methods presented in [15,18] are motivated by problems in plasticity, while the analysis in [17] uses the penalty approach in the Stokes equations to circumvent the incompressibility constraint.…”

Section: Introduction We Consider Steady Flows Of Incompressible Vismentioning

confidence: 99%

Discontinuous Galerkin finite element discretization for steady Stokes flows with threshold slip boundary condition

Djoko

2013

Quaestiones Mathematicae

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Abstract. This work is concerned with the discontinuous Galerkin finite approximations for the steady Stokes equations driven by slip boundary condition of "friction" type. Assuming that the flow region is a bounded, convex domain with a regular boundary, we formulate the problem and its discontinuous Galerkin approximations as mixed variational inequalities of the second kind with primitive variables. The well posedness of the formulated problems are established by means of a generalization of the Babuska-Brezzi theory for mixed problems. Finally, a priori error estimates using energy norm for both the velocity and pressure are obtained. (2010): 65N30, 76D07, 35J85. Mathematics Subject Classification

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Penalty finite element method for Stokes problem with nonlinear slip boundary conditions

Cited by 28 publications

References 14 publications

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

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

Discontinuous Galerkin finite element discretization for steady Stokes flows with threshold slip boundary condition

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