The AL Basis for the solution of elliptic problems in heterogeneous media
Résumé
In this paper, we will show that, for elliptic problems in heterogeneous media, there exists a local ( generalized) finite element basis (AL basis) consisting of O((log 1/H)(d+1) basis functions per nodal point such that the convergence rates of the classical finite element method for Poisson- type problems are preserved. Here H denotes the mesh width of the finite element mesh and d is the spatial dimension. We provide several numerical examples beyond our theory, where even O( 1) basis functions per nodal point are sufficient to preserve the convergence rates.