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In this paper, we build on the work of [T. Hughes, G. Sangalli, VARIATIONALMULTISCALE ANALYSIS: THE FINE-SCALE GREENS' FUNCTION, PROJECTION, OPTIMIZATION,LOCALIZATION, AND STABILIZED METHODS, SIAM Journal of Numerical Analysis,45(2), 2007] dealing with the explicit computation of the Fine-Scale Green'sfunction. The original approach chooses a set of functionals associated with aprojector to compute the Fine-Scale Green's function. The construction of thesefunctionals, however, does not generalise to arbitrary projections, higherdimensions, or Spectral Element methods.  We propose to generalise the construction of the required functionals byusing dual functions. These dual functions can be directly derived from thechosen projector and are explicitly computable. We show how to find the dualfunctions for both the $L^2$ and the $H^1_0$ projections. We then go on todemonstrate that the Fine-Scale Green's functions constructed with the dualbasis functions consistently reproduce the unresolved scales removed by theprojector.  The methodology is tested using one-dimensional Poisson andadvection-diffusion problems, as well as a two-dimensional Poisson problem. Wepresent the computed components of the Fine-Scale Green's function, and theFine-Scale Green's function itself. These results show that the method worksfor arbitrary projections, in arbitrary dimensions. Moreover, the methodologycan be applied to any Finite/Spectral Element or Isogeometric framework.

Construction and application of an algebraic dual basis and the  Fine-Scale Greens' Function for computing projections and reconstructing  unresolved scales