We present a novel enhanced finite element method for the Darcy problem starting from the non stable
continuous $P_1 / P_0$ finite element spaces enriched with multiscale functions. The method is a departure
from the standard mixed method framework used in these applications. The methods are derived in a Petrov-Galerkin
framework where both velocity and pressure trial spaces are enriched with functions based on residuals of strong
equations in each element and edge partition. The strategy leads to enhanced velocity space with an element of
the lowest order Raviart-Thomas space and to a stable weak formulation preserving local mass conservation.
Numerical tests validate the method.
Jointly with Gabriel R Barrenechea, Universidad de Concepcion &
Frederic G C Valentin, LNCC