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Abstract
In this paper we consider the NavierStokesBrinkman equations, which constitute one of the most common nonlinear models utilized to simulate viscous fluids through porous media, and propose and analyze a Banach spacesbased approach yielding new mixed finite element methods for its numerical solution. In addition to the velocity and pressure, the strain rate tensor, the vorticity, and the stress tensor are introduced as auxiliary unknowns, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation becomes a nonlinear perturbation of, in turn, a perturbed saddle point linear system, which is then rewritten as an equivalent fixedpoint equation whose operator involved maps the velocity space into itself. The wellposedness of it is then analyzed by applying the classical Banach fixed point theorem, along with a smallness assumption on the data, the BabuškaBrezzi theory in Banach spaces, and a slight variant of a recently obtained solvability result for perturbed saddle point formulations in Banach spaces as well. The resulting Galerkin scheme is momentumconservative. Its unique solvability is analyzed, under suitable hypotheses on the finite element subspaces, using a similar fixedpoint strategy as in the continuous problem. A priori error estimates are rigorously derived, including also that for the pressure. We show that PEERS and AFW elements for the stress, the velocity, and the rotation, together with piecewise polynomials of a proper degree for the strain rate tensor, yield stable discrete schemes. Then, the approximation properties of these subspaces and the Céa estimate imply the respective rates of convergence. Finally, we include two and three dimensional numerical experiments that serve to corroborate the theoretical findings, and these tests illustrate the performance of the proposed mixed finite element methods.
Original language  English 

Pages (fromto)  343373 
Number of pages  31 
Journal  Journal of Numerical Mathematics 
Volume  31 
Issue number  4 
DOIs  
Publication status  Published  5 Dec 2023 
Keywords
 a priori error analysis
 BabuškaBrezzi theory
 Banach framework
 fixedpoint theory
 mixed finite element methods
 NavierStokesBrinkman equations
 perturbed saddlepoint
Projects
 2 Active

Nextgeneration methods for transport in poroelastic media with interfaces
Australian Research Council (ARC)
1/03/23 → 28/02/27
Project: Research

Towards predictive 4D computational models for the heart
Badia, S., Ruiz Baier, R., Mardal, K. & Rodriguez Lopez, B.
9/08/22 → 8/08/25
Project: Research