Lagrange-Galerkin Discontinuous Finite Element Methods for the Navier-Stokes Equations

In this paper, Lagrange-Galerkin discontinuous finite element method is introduced for time dependent Navier-Stokes equations. The method is stable for P
_l
/P
_l
and P
_l
/P
_l-1
, l ≥ 1 combination of discontinuous discrete velocity and pressure spaces (without requiring the discrete inf-sup condition). Optimal error estimates are proved in the framework of L
²
-theory.