Local projection stabilized Lagrange–Galerkin methods for Navier–Stokes equations at high Reynolds numbers

In this article we review recent developments in the analysis and applications of local projection stabilized (LPS) Lagrange–Galerkin (LG) methods to integrate convection dominated-diffusion problems and Navier–Stokes equations at high Reynolds numbers. LG methods combine a discrete Galerkin projection method, usually finite elements, with a backward in time discretization of the convection terms along the characteristic curves. The main advantage of this approach is that provides a natural upwinding to the space discretization of the equations and transform the convection–diffusion problem into an elliptic one (or a Stokes problem in the case of Navier–Stokes equations); however, despite this upwinding introduced by the discretization of the material derivative, LG methods need to be further stabilized. The LPS technique is well suited to stabilize LG methods for the following reasons: (1) it is symmetric and does not break up the symmetry of the discrete system obtained by the backward in time discretization of the material derivative; (2) it is flexible in the sense that can be used with any conventional time marching scheme, and (3) is relatively easy to incorporate in any conventional LG code. In this article, we summarize the last theoretical results concerning the convergence and stability properties when LPS-LG methods are applied with time marching schemes of first and second order.