A SUBGRID STABILIZED METHOD FOR NAVIER-STOKES EQUATIONS WITH NONLINEAR SLIP BOUNDARY CONDITIONS

In this paper, we consider a subgrid stabilized Oseen iterative method for the Navier-Stokes equations with nonlinear slip boundary conditions and high Reynolds number. We provide one-level and two-level schemes based on this stability algorithm. The two-level schemes involve solving a subgrid stabilized nonlinear coarse mesh inequality system by applying m Oseen iterations, and a standard one-step Newton linearization problems without stabilization on the fine mesh. We analyze the stability of the proposed algorithm and provide error estimates and parameter scalings. Numerical examples are given to confirm our theoretical findings.

Parallel iterative finite-element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

Based on full domain partition, three parallel iterative finite-element algorithms are proposed and analyzed for the Navier–Stokes equations with nonlinear slip boundary conditions. Since the nonlinear slip boundary conditions include the subdifferential property, the variational formulation of these equations is variational inequalities of the second kind. In these parallel algorithms, each subproblem is defined on a global composite mesh that is fine with size h on its subdomain and coarse with size H (H ≫ h) far away from the subdomain, and then we can solve it in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Compared with the corresponding serial iterative finite-element algorithms, the parallel algorithms proposed in this paper can yield an approximate solution with a comparable accuracy and a substantial decrease in computational time. Contributions of this paper are as follows: (1) new parallel algorithms based on full domain partition are proposed for the Navier–Stokes equations with nonlinear slip boundary conditions; (2) nonlinear iterative methods are studied in the parallel algorithms; (3) new theoretical results about the stability, convergence and error estimates of the developed algorithms are obtained; (4) some numerical results are given to illustrate the promise of the developed algorithms.

Local and Parallel Finite Element Algorithms for the Stokes Equations with Nonlinear Slip Boundary Conditions

Based on two-grid discretizations, local and parallel finite element algorithms are studied for the Stokes equations with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind. Thereafter, the variational inequality can be transform into the variational identity as a multiplier in a convex set. The main idea of our algorithms is to approximate the low frequencies of the finite element solution using a coarse grid and use a fine grid to correct the resulted residual (that includes mostly high frequencies of the solution) by some local and parallel procedures. Error bounds for the approximate solutions are estimated. Numerical results are also given to demonstrate the effectiveness of the algorithms.