Uzawa iteration method for stokes type variational inequality of the second kind

Abstract

In this paper, the Uzawa iteration algorithm is applied to the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind. Firstly, the multiplier in a convex set is introduced such that the variational inequality is equivalent to the variational identity. Moreover, the solution of the variational identity satisfies the saddle-point problem of the Lagrangian functional L. Subsequently, the Uzawa algorithm is proposed to solve the solution of the saddle-point problem. We show the convergence of the algorithm and obtain the convergence rate. Finally, we give the numerical results to verify the feasibility of the Uzawa algorithm.

Two-level defect-correction stabilized finite element method for Navier–Stokes equations with friction boundary conditions

“…on S }. In this case, we can solve the problem (4) by the following Uzawa iteration method introduced in [12]:…”

“…We use the Uzawa iteration algorithm introduced by [10,12] to solve this problem, and inequality problem (4) is equivalent to the following variational equation:…”

Pressure projection stabilized finite element method for Stokes problem with nonlinear slip boundary conditions

“…Because the variational formulation (3) is variational inequality problem, if we want to give the numerical results, then problem (3) must be transformed to the variational equations. In [15], we show that the variational inequality problem (3) is equivalent to the following variational equation:…”

“…As the variational formulation of (1) and (2) is the variational inequality problem, if we want to solve this problem, then the variational inequality problem must be transformed to the variational equation. Here, we use the Uzawa iteration method in [15] by introducing the multiplier. Comparing with the usual Galerkin method without the pressure projection and the bubble method, we find that the pressure projection stabilized finite element method is valid for the Stokes problem with nonlinear slip boundary conditions (2).…”

Two-level variational multiscale finite element methods for Navier–Stokes type variational inequality problem

“…Here, we use Uzawa iteration method discussed by Y. Li and K. Li in [39], which has also been used by M. Ayadi, M. Gdoura and T. Sassi [13]. This Uzawa iteration method is based on the following equivalence relationship.…”