On a finite element approximation of the Stokes equations under a slip boundary condition of the friction type

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain Ω ⊂ ℝN (N = 2,3). We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix–Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition u · n∂Ω = g on ∂Ω. Because the original domain Ω must be approximated by a polygonal (or polyhedral) domain Ωh before applying the finite element method, we need to take into account the errors owing to the discrepancy Ω ≠ Ωh, that is, the issues of domain perturbation. In particular, the approximation of n∂Ω by n∂Ωh makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator H1 (Ω)N → H1/2(∂Ω); u ↦ u⋅n∂Ω. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates O(hα + ε) and O(h2α + ε) for the velocity in the H1- and L2-norms respectively, where α = 1 if N = 2 and α = 1/2 if N = 3. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter ϵ in the estimates.

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Finite element method for a stationary Stokes hemivariational inequality with slip boundary condition

IMA Journal of Numerical Analysis ◽

10.1093/imanum/drz032 ◽

2019 ◽

Vol 40 (4) ◽

pp. 2696-2716

Author(s):

Changjie Fang ◽

Kenneth Czuprynski ◽

Weimin Han ◽

Xiaoliang Cheng ◽

Xiaoxia Dai

Keyword(s):

Finite Element Method ◽

Boundary Condition ◽

Finite Element ◽

Numerical Solutions ◽

Stokes Equations ◽

Directional Derivative ◽

Slip Boundary Condition ◽

Hemivariational Inequality ◽

Slip Boundary ◽

Element Method

Abstract This paper is devoted to the study of a hemivariational inequality problem for the stationary Stokes equations with a nonlinear slip boundary condition. The hemivariational inequality is formulated with the use of the generalized directional derivative and generalized gradient in the sense of Clarke. We provide an existence and uniqueness result for the hemivariational inequality. Then we apply the finite element method to solve the hemivariational inequality. The incompressibility constraint is treated through a mixed formulation. Error estimates are derived for numerical solutions. Numerical simulation results are reported to illustrate the theoretically predicted convergence orders.

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Finite element analysis of the stationary power-law Stokes equations driven by friction boundary conditions

Journal of Numerical Mathematics ◽

10.1515/jnma-2015-0003 ◽

2015 ◽

Vol 23 (1) ◽

Cited By ~ 4

Author(s):

Jules K. Djoko ◽

Mohamed Mbehou

Keyword(s):

Finite Element ◽

Boundary Conditions ◽

Power Law ◽

Stokes Equations ◽

Finite Element Approximation ◽

Element Approximation ◽

Mixed Variational Inequality ◽

Element Analysis ◽

Slip Boundary ◽

Nonlinear Slip Boundary Conditions

AbstractIn this work, we are concerned with the finite element approximation for the stationary power law Stokes equations driven by nonlinear slip boundary conditions of ‘friction type’. After the formulation of the problem as mixed variational inequality of second kind, it is shown by application of a variant of Babuska-Brezzi’s theory for mixed problems that convergence of the finite element approximation is achieved with classical assumptions on the regularity of the weak solution. Next, solution algorithm for the mixed variational problem is presented and analyzed in details. Finally, numerical simulations that validate the theoretical findings are exhibited.

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Involving the Navier–Stokes equations in the derivation of boundary conditions for the lattice Boltzmann method

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2011.0045 ◽

2011 ◽

Vol 369 (1944) ◽

pp. 2184-2192 ◽

Cited By ~ 1

Author(s):

Joris C. G. Verschaeve

Keyword(s):

Boundary Condition ◽

Lattice Boltzmann Method ◽

Lattice Boltzmann ◽

Stokes Equations ◽

Slip Boundary Condition ◽

Navier Stokes ◽

Grid Spacing ◽

Navier Stokes Equations ◽

Slip Boundary ◽

Boltzmann Method

By means of the continuity equation of the incompressible Navier–Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.

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