Spectral discretizations of the Darcy's equations with non standard boundary conditions

This paper is devoted to the approximation of anonstandard Darcy problem, which modelizes the flow in porous media, byspectral methods: the pressure is assigned on a part of the boundary.We propose two variational formulations, as well as three spectraldiscretizations. The second discretization improves the approximation of thedivergence-free condition, but the error estimate on the pressure is notoptimal, while the third one leads to optimal error estimate with adivergence-free discrete solution, which is important for someapplications. Next, their numerical analysis is performed in detailand we present some numerical experiments which confirm the interestof the third discretization.