Nonlinear Galerkin approximation of the two dimensional exterior Navier-Stokes problem

In this paper, we present an Oseen coupling problem to approximate the two dimensional exterior unsteady Navier-Stokes problem with the nonhomogeneous boundary conditions. The Oseen coupling problem consists of the Navier-Stokes equations in a bounded region and the Oseen equations in an unbounded region. Then we derive the reduced Oseen coupling problem by use of the integral representations of the solution of the Oseen equations in an unbounded region. Moreover, we present the Galerkin approximation and the nonlinear Galerkin approximation for the reduced Oseen coupling problem. By analysing their convergence rates, we find that the nonlinear Galerkin approximation provides the same convergence order as the classical Galerkin approximation if we choose the space discrete parameter $H=O(h^{1/2})$. However, in this approximation, the nonlinearity is treated on the coarse grid finite element space and only the linear problem needs to be solved on the fine grid finite element space.