On the relationship of local projection stabilization to other stabilized methods for one-dimensional advection–diffusion equations

We consider the one-level approach of the local projection stabilization (LPS) for solving a singularly perturbed advection–diffusion two-point boundary value problem. Eliminating the enrichments we end up with the differentiated residual method (DRM) which coincides for piecewise linears with the streamline upwind Petrov–Galerkin (SUPG) method and for piecewise polynomials of degree r ⩾ 2 with the variational multiscale method (VMS). Furthermore, we show that in certain cases the stabilization parameter can be chosen in such a way that the piecewise linear part of the solution becomes nodal exact. In this way, we obtain explicit formulas for the stabilization parameter depending on the local meshsize, the polynomial degree r of the approximation space, and the data of the problem. Finally, we discuss the behaviour of different modes of the discrete solution when varying the stabilization parameter.