A multi-level discontinuous Galerkin method for solving the stationary Navier–Stokes equations

In this paper, we consider the multi-level discontinuous finite element method for solving the stationary incompressible Navier–Stokes equations. On the coarsest mesh the discrete nonlinear Navier–Stokes equations are solved by using piecewise polynomial functions, which are totally discontinuous across inter-element boundaries and are pointwise divergence free on each element for the velocity and are continuous functions for the pressure. Subsequent approximations are generated on a succession of refined grids by solving the Newton linearized Navier–Stokes equations using piecewise polynomial functions which are similar to that on the coarsest mesh. Finally, the well-posedness and the optimal error estimate for the multi-level discontinuous Galerkin method are provided. The error analysis shows that when the mesh scales kj+1=O(kj2),hj+1=O(hj2) with j=0,1,…,J−1j=0,1,…,J−1 are chosen, the multi-level finite element method can save a large amount of computational time compared with the one-level finite element method.