Adaptive Local (AL) Basis for Elliptic Problems with $L^\infty$-Coefficients

TL;DR

This paper introduces an adaptive local finite element basis method for elliptic PDEs with heterogeneous media, achieving optimal convergence rates while using a limited number of basis functions per mesh point.

Contribution

It presents a novel adaptive local basis construction that does not require resolving media heterogeneity at the mesh level, maintaining optimal convergence.

Findings

Requires O(log(1/H)^{d+1}) basis functions per mesh point.

Preserves optimal finite element convergence rates.

Applicable to elliptic problems with $L^ abla$-coefficients in heterogeneous media.

Abstract

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.