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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.