On the Stokes Equation with the Leak and Slip Boundary Conditions of Friction Type: Regularity of Solutions

Saito, Norikazu

doi:10.2977/prims/1145475807

Cited by 114 publications

(102 citation statements)

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“…in [6], [7], [15]. The main goal of this paper is to study under which conditions concerning the smoothness of Ω solutions to the Stokes problem with threshold slip depend continuously on variations of Ω.…”

Section: Introductionmentioning

confidence: 99%

Shape optimization for Stokes problem with threshold slip

2014

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Section: Introductionmentioning

confidence: 99%

Shape optimization for Stokes problem with threshold slip

2014

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“…Subsequently, many studies have focused on the properties of the solution of the resulting boundary value problem, for example, existence, uniqueness, regularity, and continuous dependence on data, for Stokes, Navier-Stokes and Brinkman-Forchheimer equations under such boundary conditions. Details can be found in [1,3,11,12,13,14,15,16,17,43,44,45,46] among others.…”

Section: Preliminaries and Weak Formulationsmentioning

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 unique solvability and regularity of the continuous problem (1.8) has been discussed in detail for the different boundary conditions for instance in [40], [38], [34] and [29]. The theory therein bases mainly on Korn's second inequality and its variants (cf.…”

Section: Uniqueness In the Pressure Variable Of The Solution Can Onlymentioning

confidence: 99%

Finite element methods for the Stokes problem on complicated domains

Peterseim

Sauter

2011

Computer Methods in Applied Mechanics and Engineering

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Abstract. It is a standard assumption in the error analysis of finite element methods that the underlying finite element mesh has to resolve the physical domain of the modeled process. In case of complicated domains appearing in many applications such as ground water flows this requirement sometimes becomes a bottleneck. The resolution condition links the computational complexity a priorily to the number (and size) of geometric details. Therefore even the coarsest available discretization can lead to a huge number of unknowns. In this paper, we will relax the resolution condition and introduce coarse (optimal order) approximation spaces for Stokes problems on complex domains. The described method will be efficient in the sense that the number of unknowns is only linked to the properties of the solution and not to the problem data. The presentation picks up the concept of composite finite elements for the Stokes problem presented in a previous paper of the authors. Here, the a priori error and stability analysis of the proposed mixed method is generalized to quite general, i.e. slip and leak boundary conditions that are of great importance in practical applications. Keywords Problem settingWe consider the stationary Stokes equationsdescribing the motion of a viscous incompressible fluid in a bounded Lipschitz domain Ω under the general mixed boundary conditions proposed by Navier [25] Thereby we use the following notationBoth, the equations (1.1) and the boundary conditions (1.2), can be generalized by replacing the zeros on the right hand sides by some given functions. In this paper, we are especially interested in the limit cases of the boundary conditions, i.e., Dirichlet (λ ν = λ τ = 1), Neumann (λ ν = λ τ = 0), slip (λ ν = 1, λ τ = 0) and leak (λ ν = 0, λ τ = 1) boundary conditions. In particular, we assume the boundary Γ := ∂ Ω to consist of four relatively closed disjoint partsThe leak and slip parts Γ s and Γ l are supposed to be of class C 1 . We define the coefficient functions from (1.2) in the following way:This choice leads to the following set of boundary conditionsWhile the mathematical literature on Dirichlet and Neumann boundary conditions is vast, leak and slip boundary conditions have been studied less extensively. However, they are of great practical interest. For theoretical studies of these boundary conditions we refer to [38; 13; 34] and for some applications to [19; 20].The Sobolev space that contains those velocity fields which fulfill the essential parts (conditions on the left in (1.6)) of the boundary conditions is denoted byThe (mixed) weak formulation of problem (1.1) together with the boundary conditions (1.6) reads:The bilinear forms a :In general, problem (1.8) is not uniquely solvable. The bilinear form a has a nontrivial kernel given by the set of rigid body motionsMoreover, every v ∈ R is divergence-free, i.e. the pairs (v, q), q ∈ R, is a solution of the homogeneous Stokes problem. Thus, a solution of (1.8) can only be unique up to elements of H 1 ess ∩ R. Remark 1.To assure ...

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On the Stokes Equation with the Leak and Slip Boundary Conditions of Friction Type: Regularity of Solutions

Cited by 114 publications

References 20 publications

Shape optimization for Stokes problem with threshold slip

Shape optimization for Stokes problem with threshold slip

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

Finite element methods for the Stokes problem on complicated domains

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