A Petrov–Galerkin enriched method: A mass conservative finite element method for the Darcy equation

Starting from the non-stable P1/P0P1/P0 discretization we build enhanced methods for the Darcy equation which are stable and locally mass-conservative. The methods are derived in a Petrov–Galerkin framework where both velocity and pressure trial spaces are enriched with multiscale functions. These functions solve local problems correcting the residuals of the strong equations in each element and interior edge, which leads to a velocity space enhanced with functions belonging to the lowest order Raviart–Thomas space. Several numerical tests validate the methods.