What question did this study set out to answer?

The aim is to construct a local generalized finite element basis for elliptic problems with heterogeneous coefficients.

Localization of elliptic multiscale problems

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Demonstrates error decay in elliptic problems through localized finite element bases, indicating improvements in computational efficiency.

The aim is to construct a local generalized finite element basis for elliptic problems with heterogeneous coefficients.
Development of basis functions as solutions to local problems on vertex patches.
Analysis of error decay rate in the generalized finite element method based on the number of element layers in patches.
Investigation of patch sizes needed for a linear rate of convergence.
Error decays exponentially with respect to the number of layers of elements in the patches.
Patches with diameter H log(1/H) maintain a linear convergence rate in H.
Analysis does not depend on solution regularity or scale separation in coefficients.

Målqvist et al. (2014) studied this question.

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Demonstrates error decay in elliptic problems through localized finite element bases, indicating improvements in computational efficiency.