Adaptive Local (AL) Basis for Elliptic Problems with $L^\infty$-Coefficients
Provides a rigorous method for coarse-mesh discretization of heterogeneous media problems, preserving optimal convergence rates.
The paper introduces an adaptive local finite element basis for elliptic PDEs with highly heterogeneous coefficients, achieving optimal convergence rates with O(log(1/H)^{d+1}) basis functions per mesh point.
We define a generalized finite element method for the discretization of elliptic partial differential equations in heterogeneous media. An adaptive local finite element basis (AL basis) on a coarse mesh which does not resolve the matrix of the media is constructed by solving finite-dimensional localized problems. The method requires $O(log(1/H)^{d+1})$ basis functions per mesh point. We prove that the optimal finite element convergence rates are preserved.