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This paper addresses a multi-scale finite element method for second orderlinear elliptic equations with arbitrarily rough coefficient. We propose alocal oversampling method to construct basis functions that have optimal localapproximation property. Our methodology is based on the compactness of thesolution operator restricted on local regions of the spatial domain, and doesnot depend on any scale-separation or periodicity assumption of thecoefficient. We focus on a special type of basis functions that are harmonic oneach element and have optimal approximation property. We first reduce ourproblem to approximating the trace of the solution space on each edge of theunderlying mesh, and then achieve this goal through the singular valuedecomposition of an oversampling operator. Rigorous error estimates can beobtained through thresholding in constructing the basis functions. Numericalresults for several problems with multiple spatial scales and high contrastinclusions are presented to demonstrate the compactness of the local solutionspace and the capacity of our method in identifying and exploiting this compactstructure to achieve computational savings.

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