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

Abstract

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.

Low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions

“…Kashiwabara studied discrete variational inequality problem for the Stokes equations with leak boundary conditions of friction type. Other theoretical and numerical results for the steady and unsteady Stokes/Navier‐Stokes problems with such boundary conditions can be found in previous studies …”

“…Moreover, if ∇· u =0, then trilinear form b (·,·,·) satisfies which has the following properties: where N is a positive constant depending only on Ω.…”

“…Other theoretical and numerical results for the steady and unsteady Stokes/Navier-Stokes problems with such boundary conditions can be found in previous studies. [15][16][17][18][19][20][21] The development of an efficient finite element method for the incompressible flow simulations is an important but challenging problem. The importance of ensuring the compatibility of the component approximations of velocity and pressure by satisfying the discrete inf-sup condition is widely known.…”

Two‐grid variational multiscale algorithms for the stationary incompressible Navier‐Stokes equations with friction boundary conditions

“…These theoretical problems were also studied by many scholars, such as Fujita et al , Saito et al , Li et al , and references cited therein. Other numerical theory results about steady and time‐dependent Navier‐Stokes problems with friction boundary conditions can be found in .…”

Fully Discrete Finite Element Methods for Two-Dimensional Bingham Flows

“…Generally, there exist two main numerical techniques: one based on the regularized method [1] and the other based on the augmented Lagrangian method [2]. Because the regularized method makes variational inequality problems into the equations, it is a powerful tool to deal with the variational inequality problems, such as the obstacle problem [3,4], Navier-Stokes problems with subdifferential boundary conditions [5,6], and etc.…”