New degrees of freedom for high-order Whitney approximations of Darcy’s flows

Abstract

We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. We propose a high-order discretisation based on Whitney finite elements, namely, Raviart-Thomas finite elements of degree r + 1 for the discharge and discontinuous piecewise polynomial finite elements of degree r for the pressure, with r ≥ 0. We comment on the use of new degrees of freedom that have a clear physical meaning, the so-called weights on the small simplices, for the involved discharge and pressure fields. We describe a new numerical strategy to solve the discrete problem based on a tree-cotree block-decomposition of the unknowns that is natural when considering these new degrees of freedom. Preliminary numerical tests in two dimensions confirm the stability of the adopted method and the effectiveness of the new degrees of freedom.