On fully discrete Galerkin approximations for the incompressible Navier-Stokes equations

Fully discrete approximations to the solution of the Incompressible Navier-Stokes equations are introduced and analyzed. Implicit Runge-Kutta methods are used for the temporal discretizations. Standard elements are used for the pressure approximation, while non-conforming finite element spaces are used for the velocity approximation. These elements are discontinuous across interelement boundaries and satisfy the incompressibility condition pointwise on each "trian-gle". Furthermore, no global quasi-uniformity condition is required from the sub-divisions of the domain. Newton's method and a more efficient Implicit-Explicit scheme are employed to solve the resulting system of nonlinear equations. Numer-ical solutions to two well known physical benchmark problems are presented.