We apply the Streamline Upwind Petrov Galerkin (SUPG) finite element formulation of the Level Set method to 2D redistancing and advection problems on unstructured triangulated grids. The purpose is to test the Level Set method for mass conservation properties, where the mass is understood as the amount of fluid enclosed by the interface. For the redistancing procedure we implement the idea of mass correction suggested by Sussman and Fatemi (1999) and confirm its high accuracy within the finite element formulation. However, we find that the use of the first order SUPG formulation of the Level Set method for coupled redistancing-advection problems can result in significant loss of mass caused by distortion of the interface due to numerical diffusion. This neccesitates the use of higher order upwind finite element schemes for the advection equation.