Optimal Local Multi-scale Basis Functions for Linear Elliptic Equations with Rough Coefficient

TL;DR

This paper introduces a local multi-scale finite element method for linear elliptic equations with rough coefficients, using oversampling basis functions that do not rely on scale separation, achieving optimal approximation and computational efficiency.

Contribution

It develops a novel oversampling basis function construction based on solution operator compactness, applicable to problems with arbitrary rough coefficients without scale separation.

Findings

Numerical results demonstrate effective approximation for multiscale problems.

The method achieves computational savings by exploiting local solution space compactness.

Error estimates confirm the optimal approximation properties of the basis functions.

Abstract

This paper addresses a multi-scale finite element method for second order linear elliptic equations with arbitrarily rough coefficient. We propose a local oversampling method to construct basis functions that have optimal local approximation property. Our methodology is based on the compactness of the solution operator restricted on local regions of the spatial domain, and does not depend on any scale-separation or periodicity assumption of the coefficient. We focus on a special type of basis functions that are harmonic on each element and have optimal approximation property. We first reduce our problem to approximating the trace of the solution space on each edge of the underlying mesh, and then achieve this goal through the singular value decomposition of an oversampling operator. Rigorous error estimates can be obtained through thresholding in constructing the basis functions. Numerical results for several problems with multiple spatial scales and high contrast inclusions are presented to demonstrate the compactness of the local solution space and the capacity of our method in identifying and exploiting this compact structure to achieve computational savings.