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

Abstract

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

Analysis of a Time Implicit Scheme for the Oseen Model Driven by Nonlinear Slip Boundary Conditions

“…because J is lower semi-continuous, one can pass to the limit in (2.20) (see [1,19]) and obtain the following result Theorem 3.1. Given f ∈ L 2 (0, T ; H), g ∈ L 2 (S), and u 0 ∈ H, there exists u satisfying; 17) and such that ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u(0) = u 0 in Ω, and for all v ∈ V div , and a.e t ∈ (0, T )…”

“…In this work instead we will follow the approach advocate in [6,7] which control the consistency error by introducing discrete dissipative (non negative) quantities and careful utilization of sub-differential operator. To the best of our knowledge, almost all works in the literature for fluids flow driven by nonlinear slip boundary conditions of friction type are concerned with existence of solutions or its finite element discretizations (see [8][9][10][11][12][13][14][15][16][17] just to mention a few). A fully discrete approximation of the Navier Stokes system with Tresca boundary condition has been considered in Li and Li in [8], but in order to obtain convergence for the discretization in time higher regularity of the exact solution is needed.…”

“…, N, one obtainsN n=1 u(t) − u k (t) 2 ≤ 2k 2 νC K C P ν (0,T ;L 2 ) + 2k 2 νC K C P N n=1 E n + 4 νC K C P f k − f 2 L 2 (0,T ;L 2 ) ,which leads to (4.10) after application of Corollary 4.1, and error estimatef k n − f L 2 (tn−1,tn;L 2 ) ≤ Ck f L 2 (tn−1,tn;L 2 ) . (4.16)Next, integrating (4.14) over (t n−1 , t n ), using (2.8), one obtainstn tn−(u − u k ) 2 ds ≤ C u(t n−1 ) − u k (t n−1 ) 2 + Ck 3 ∇δu k n − f (s) 2 ds + k 2 E n ,(4 17).…”

Discontinuous Galerkin methods for solving a hyperbolic inequality

“…Moreover, locality of the discretization makes the DG methods ideally suited for parallel computing (see and the references therein). Recently, DG methods have been applied for solving VIs, such as gradient plasticity problem , obstacle problems , Signorini problem , quasistatic contact problems , plate contact problem , two membranes problem and Stokes or Navier–Stokes flows with slip boundary condition . A posteriori error analysis of DG methods for VIs was also considered in .…”

Convergence Analysis of the Nonconforming Finite Element Discretization of Stokes and Navier–Stokes Equations with Nonlinear Slip Boundary Conditions

“…The main concern in this research is to analyze numerically problem (P) and problem (F) using the nonconforming finite element method where the velocity is approximated by lowest order Crouzeix-Raviart element and the pressure with piecewise constant functions [10]. The a priori error analysis of problem (P) has been proposed with discontinuous Galerkin method in [11], while numerous studies using conforming approximation of the velocity have been contributed by researchers, see among others [12,13,14,15,16,17,18]. The finite element method is now well adapted for approximating the solution of partial differential equations written in weak form (including variational inequalities, see [19,20,21,22,23,24]), and the search for efficient and simple non conforming finite element methods for Stokes, Navier-Stokes equations driven by nonlinear slip boundary conditions has not yet been well explored by researchers (except the early work of the author in [11]).…”