Implicit scheme for Navier-Stokes equations

Abstract

Navier-Stokes equations have convective terms which make the equations nonlinear. According to the method computing the convective terms, there are many different schemes. Among them implicit schemes are more stable than explicit ones. Especially when the Reynolds number is large. In this paper, we introduce an implicit scheme for stationary incompressible viscous Navier-Stokes equations. We check its convergence and the error estimate. We see that the regularity of the solution of Navier-Stokes equations is important to get numerically a good approximate solution of the variational form. For the space discretization, we employ the standard mixed finite element formulation and use the Hood-Taylor element. The driven flow in a square cavity is used as the model problem. Solutions are obtained for Reynolds number 1000 and 2000. For the division of the domain, each side is divided into 15 non-uniform intervals. The total number of elements (of p) is 450.