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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.

In this paper, we give a new mixed variational formulation to the Poisson equation based on the less regularity of f lux(velocity) in practice, and show the existence and uniqueness of the solution to this saddle point problem. Based on this new formulation, we address its corresponding stabilization conforming the finite-element approximation for P 2 1 − P 1 finite-element pairs based on two local Gauss integrations for velocity, and give the finite-element solution's existence and uniqueness. Moreover, we obtain that the approximation of pressure p is optimal in H 1 -and L 2 -norms, the approximation of velocity u is suboptimal in H 1 -norm. Finally, we give some numerical experiment to verify the theoretical results.

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The stationary Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper. Because the boundary conditions include the subdifferential property on the part boundary, the variational formulation of this problem is the variational inequality problem of the second kind with Navier-Stokes operator. The main purpose of the paper is to study the existence of the weak solution and the strong solution to this variational inequality problem in terms of the Yosida's regularity method.

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gâteaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.

SUMMARYIn this paper, by using classical tensor calculus, we derive the compressible Navier-Stokes equation on a so-called stream surface which is a two-dimensional (2-D) manifold that gives a deÿnition of a stream function with the equation satisÿed by it. Based on this, a new algorithm is proposed which is called dimension split algorithm. This new method is di erent from the domain decomposition method. In the domain decomposition method, we have to solve a three-dimensional problem in each subdomain but we solve a 2-D problem in each subdomain. A numerical experiment for the turbomachine is provided.