I was reading about Petrov-Galerkin Enrichment Method for Darcy equations. Here are a couple papers that discuss this in detail:

However, I am not sure I understand what they are doing completely and would like some help understanding how they are constructing these "multiscale" or enrichment functions.

From what I understand, it seems they are "enriching" the $P_1$ velocity by an $RT_0$, but what exactly does this mean? Specifically, what is meant when they "correct edge residuals"? Because I would like to implement this in something like FEniCS (if possible) but in order to do that I need to understand this method completely.



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