Optimal error Estimates for the Stokes and Navier–Stokes equations with slip–boundary condition

Abstract

Abstract. We consider a finite element discretization by the Taylor-Hood element for the stationary Stokes and Navier-Stokes equations with slip boundary condition. The slip boundary condition is enforced pointwise for nodal values of the velocity in boundary nodes. We prove optimal error estimates in the H 1 and L 2 norms for the velocity and pressure respectively.

A Lagrangian–Eulerian approach for the numerical simulation of free-surface flow of a viscoelastic material

“…However, Eq. (13) gives an explicit formula for σ n in terms of already calculated quantities and of u n :…”

Finite element discretization of the Navier–Stokes equations with a free capillary surface

“…in [2]. In [2,4] one can also find a simple and efficient way how to deal with such a boundary condition from an algorithmical point of view.…”

Penalty Method for the Stationary Navier–Stokes Problems Under the Slip Boundary Condition

“…In [21,22], the P1/P1 element with pressure stabilization is used, and the slip boundary condition is implemented as u h ( p) · n( p) = 0 for all vertices p of Γ h , where u h is the approximate solution. A similar method presented in [3] is to introduce a homeomorphism G h : Ω h → Ω and to consider the P2/P1 element, then the slip boundary condition is described as u h ( p) · n(G h ( p)) = 0, where p is either a vertex or a midpoint of edges of Γ h . In those papers, the spherical domain Ω is considered, where n(x) is easy to obtain; however, for a general domain, it is more convenient to use n h than n in finite element approximation.…”

“…[3,21]) to avoid the variational crime, there exists a penalty method (see [7,10]) to approximate the slip boundary condition. The penalty method has several advantages: it facilitates the numerical implementation since only n h is involved in FEM, and it also enables us to avoid the variational crime, as explained below.…”