A New Divergence-Free Interpolation Operator with Applications to the Darcy--Stokes--Brinkman Equations

A local interpolation operator, preserving the divergence, is constructed explicitly, which is the first ever one for the divergence-free elements. A divergence-free finite element method is applied to the Darcy-Stokes-Brinkman flow in a mixed region of both free and porous medium. The method is of optimal order in contrast to the traditional H and H mixed finite elements, which work for either the Darcy flow or the Stokes flow but not the mixture. Comparing to the existing non-conforming elements, the divergence-free element method provides a continuous solution for the velocity which is also an orthogonal projection within a Hilbert subspace of the true velocity. Numerical tests supporting the theory are presented.