A stabilized numerical method for the Darcy–Brinkman–Forchheimer equations
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We present a stabilized finite element method for the Darcy–Brinkman–Forchheimer equations which we derived within the variational multiscale (VMS) paradigm. In particular, we consider both the ASGS and the OSS variants of the method, which differ in the expression of the projection operator that is applied to the residual of the equations to derive the stabilized system. Our analysis and numerical experiments show that the properties of the resulting methods are very similar to those observed in the standard Navier-Stokes system. We also look at the error introduced by interpolating the exact (given) porosity field by finite element functions, which is expected to be of relevance in practice. Our results indicate that such interpolation error does not deteriorate the overall convergence properties if certain restrictions on the ranges of the physical parameters involved are fulfilled. An important motivation behind this work is to move toward building a firm basis for the development of CFD-DEM algorithms that use a finite element discretization of the continuous phase.