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

Abstract

This paper is devoted to the approximation of a nonstandard Darcy problem, which modelizes the flow in porous media, by spectral methods: the pressure is assigned on a part of the boundary. We propose two variational formulations, as well as three spectral discretizations. The second discretization improves the approximation of the divergence-free condition, but the error estimate on the pressure is not optimal, while the third one leads to optimal error estimate with a divergence-free discrete solution, which is important for some applications. Next, their numerical analysis is performed in detail and we present some numerical experiments which confirm the interest of the third discretization.

Keywords

• Spectral method
• Darcy's equations
• numerical analysis