Non-overlapping Schwarz algorithms for the Incompressible Navier-Stokes equations with DDFV discretizations

We propose and analyze non-overlapping Schwarz algorithms for the domain decomposition of the unsteady incompressible Navier-Stokes problem with Discrete Duality Finite Volume discretizations. The design of suitable transmission conditions for the velocity and the pressure is a crucial issue. We establish the well-posedness of the method and the convergence of the iterative process, pointing out how the numerical fluxes influence the asymptotic problem which is intended to be a discretization of the Navier-Stokes equations on the entire computational domain.Finally we numerically illustrate the behavior and performances of the algorithm. We discuss on numerical grounds the impact of the parameters for several mesh geometries and we perform simulations of the flow past an obstacle with several domains.

AN OVERLAPPING SCHWARZ METHOD FOR SINGULARLY PERTURBED FOURTH-ORDER CONVECTION-DIFFUSION TYPE

In this paper, we have constructed an iterative numerical method based on an overlapping Schwarz procedure with uniform mesh for singularly perturbed fourth-order of convection-diffusion type. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the central finite difference scheme on a uniform mesh while on the non-layer region we use the mid-point difference scheme on a uniform mesh. It is shown that the method produces numerical approximations which converge in the maximum norm to the exact solution. We prove that, when appropriate subdomains are used the method produces convergence of almost second-order. Furthermore, it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results.