The AL Basis for the Solution of Elliptic Problems in Heterogeneous Media
Lars Grasedyck, Isabelle Greff, Stefan A Sauter · Multiscale Modeling and Simulation · 2012
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 $O\big( \big( \log\frac{1}{H}\big) ^{d+1}\big)$ 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.