Layer-adapted meshes for one-dimensional reaction–convection–diffusion problems

We study convergence properties of an upwinded finite element method for the solution of linear one-dimensional reaction–convection–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for (almost) first-order convergence in the L ∞ norm, uniformly in the diffusion parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of convergence. The key ingredients of our analysis are sharp bounds on the W 1,1 norm of the discrete Green's function associated with the discretization.