Gabriel N. Gatica, Antonio Marquez, Ricardo Oyarzua, Ramiro Rebolledo:
Analysis of an augmented fully-mixed approach for the coupling of quasi-Newtonian fluids and porous media
In this paper we introduce and analyze an augmented mixed finite element method for the coupling of quasi-Newtonian fluids and porous media. The flows are governed by a class of nonlinear Stokes and linear Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We apply dual-mixed formulations in both domains, and, in order to handle the nonlinearity involved in the Stokes region, we set the strain and vorticity tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly, which yields the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. Moreover, in order to facilitate the analysis, we augment the formulation in the fluid by incorporating a redundant Galerkin-type term arising from the quasi-Newtonian constitutive law multiplied by a suitable stabilization parameter. In this way, under a suitable and explicit choice of this parameter, a generalization of the Babuska-Brezzi theory is utilized to show the well-posedness of the continuous and discrete formulations and to derive the corresponding a-priori error estimate. In particular, the feasible finite element subspaces include PEERS and Arnold-Falk-Winther elements for the stress, velocity and vorticity in the fluid, Raviart-Thomas elements and piecewise constants for the velocity and pressure in the porous medium, together with piecewise constant Stokes strain tensor and continuous piecewise linear elements for the traces. Next, we employ classical approaches, which include linearization techniques, the Clement interpolator and the Helmholtz decomposition, together with known efficiency estimates, to derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. Finally, several numerical results confirming the good performance of the method and the properties of the a posteriori error estimator, and illustrating the capability of the corresponding adaptive algorithm to identify the singular regions of the solution, are reported.
This preprint gave rise to the following definitive publication(s):
Gabriel N. GATICA, Antonio MARQUEZ, Ricardo OYARZUA, Ramiro REBOLLEDO: Analysis of an augmented fully-mixed approach for the coupling of quasi-Newtonian fluids and porous media. Computer Methods in Applied Mechanics and Engineering, vol. 270, 1, pp. 76-112, (2014).