Yueqiang Shang

Newton Iterative Parallel Finite Element Algorithm for the Steady Navier-Stokes Equations

A combination method of the Newton iteration and parallel finite element algorithm is applied for solving the steady Navier-Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the Newton iterations of m times for a nonlinear problem on a coarse grid in domain Ω and computing a linear problem on a fine grid in some subdomains Ωj⊂Ω with j=1,…,M in a parallel environment. Then, the error estimation of the Newton iterative parallel finite element solution to the solution of the steady Navier-Stokes equations is analyzed for the large m and small H and h≪H. Finally, some numerical tests are made to demonstrate the the effectiveness of this algorithm.

Fourier analysis of Schwarz domain decomposition methods for the biharmonic equation

Schwarz methods are an important type of domain decomposition methods. Using the Fourier transform, we derive error propagation matrices and their spectral radii of the classical Schwarz alternating method and the additive Schwarz method for the biharmonic equation in this paper. We prove the convergence of the Schwarz methods from a new point of view, and provide detailed information about the convergence speeds and their dependence on the overlapping size of subdomains. The obtained results are independent of any unknown constant and discretization method, showing that the Schwarz alternating method converges twice as quickly as the additive Schwarz method.

ON THE LINEARIZATION OF DEFECT-CORRECTION METHOD FOR THE STEADY NAVIER-STOKES EQUATIONS

Based on finite element discretization, two linearization approaches to the defect-correction method for the steady incompressible Navier-Stokes equations are discussed and investigated. By applying times of Newton and Picard iterations to solve an artificial viscosity stabilized nonlinear Navier-Stokes problem, respectively, and then correcting the solution by solving a linear problem, two linearized defect-correction algorithms are proposed and analyzed. Error estimates with respect to the mesh size , the kinematic viscosity , the stability factor and the number of nonlinear iterations for the discrete solution are derived for the linearized one-step defect-correction algorithms. Efficient stopping criteria for the nonlinear iterations are derived. The influence of the linearizations on the accuracy of the approximate solutions are also investigated. Finally, numerical experiments on a problem with known analytical solution, the lid-driven cavity flow, and the flow over a backward-facing step are performed to verify the theoretical results and demonstrate the effectiveness of the proposed defect-correction algorithms.