Parallel Finite Element Algorithms For The Incompressible Flows With Nonlinear Slip Boundary Conditions

Stokes equations and Navier-Stokes equations are equations that reflect the basic mechanical laws of viscous incompressible fluid flow,which paly an important role in Computational Fluid Dynamics.For some flow phenomena in environmental and medical problems,such as the flow of oil layers above or below the sand layer,blood flow in the thoracic aorta and flow through a canal or a drainpipe with its bottom covered by sherbet of mud and pebbles,traditional homogeneous Dirichlet boundary conditions are no longer applicable,so the study for incompressible flows with nonlinear slip boundary conditions are of great significance.Based on full domain partition technique,some parallel finite element algorithms for the Stokes and the Navier-Stokes equations with nonlinear slip boundary conditions are designed and analyzed.The variational formulations of these equations are variational inequalities of the second kind due to the subdifferentional property included by the nonlinear slip boundary conditions.Each subproblem in parallel algorithms is solved on a global composite mesh with the vast majority of the degrees of freedom associated with the particular subdomain that it is responsible for and hence can be solved in parallel with other subproblems by using an existing sequential solver without extensive recoding.All of the subproblems for Navier-Stokes equations are nonlinear and are independently solved by three kinds of iterative methods.We can estimate the optimal error bounds of the approximate solutions with the use of some(strong)uniqueness conditions.Numerical results are also given to demonstrate the effectiveness of the parallel algorithms.