Sufficient conditions for uniform convergence on layer-adapted meshes for one-dimensional reaction–diffusion problems

Abstract

We study convergence properties of a finite element method with lumping for the solution of linear one-dimensional reaction–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for 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 estimates for the discrete Green function associated with the discretization.