A model for incompressible fluids using finite element methods for the Brinkman problem

Abstract

Abstract The partial differential equations for fluid flow dynamics based on the Brinkman equations, written in terms of velocity-vorticity and pressure in three dimensions, are essential for predicting climate, ocean currents, water flow in a pipe, the study of blood flow and any phenomenon involving incompressible fluids through porous media; having a significant impact in areas such as oceanographic engineering and biomedical sciences. This paper aims to study the Brinkman equations with homogeneous Dirichlet boundary are studied, the existence and uniqueness of solution at a continuous level through equivalence of problems is presented. It is discretized to approximate the solution using Nédélec finite elements and piecewise continuous polynomials to approximate vorticity and pressure. The velocity field is recovered, obtaining its a priori error estimation and order of convergence. As a result, ensuring a single prediction of the flow behavior of an incompressible fluid through porous media. Finally, a numerical example in 2D with the standard L2 is presented, confirming the theoretical analysis.