Abstract
The Generalized Finite Element Methods (GFEM) are known for accurately resolving local features in heterogeneous media. Recent numerical experiments have shown that the use of local bases made from A-harmonic extensions of well chosen boundary data have nearly optimal approximation properties. In this paper we show that the explanation is geometric and given by the angle between any finite dimensional approximation space and the optimal finite dimensional approximation space introduced in (Babǔska and Lipton, 2011). The effect of the angle is quantified by an upper bound on the convergence rate. This rate is seen in the numerical simulations where local bases with nearly optimal approximation properties are identified.
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