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In this work, we introduce a new locally conservative, low-order $\mathbb{P}_1\times \mathbb{P}_0$ stabilized finite element method for the Brinkman problem. The approach relies on unusual stabilizing terms (Barrenechea and Valentin, 2002), combined with additional pressure jump stabilization on mesh edges. Both stabilization mechanisms are designed to handle highly contrasting permeability coefficients across different regions of the porous medium (e.g., the rock matrix and vuggy zones), as well as boundary layer effects. In addition, we propose a straightforward post-processing procedure for the velocity field that ensures exact mass conservation at the element level. Optimal convergence rates are verified numerically using a manufactured analytical solution in the natural norms. To assess the performance of the method in highly contrasted media, we solve the Brinkman model in a porous domain containing a vuggy inclusion, using permeability values of $1\,m^2$ and $10^{-10} m^2$ for the vug and the rock matrix, respectively. The results are compared with a reference solution computed on a highly refined mesh using Taylor–Hood finite elements, showing good agreement for both pressure and velocity fields across various mesh resolutions. For both the analytical and the vuggy test cases, we verify that the post-processed velocity field satisfies the mass conservation equation to machine precision. As a result, the proposed low-order finite element method provides an affordable and accurate alternative for computing velocity and pressure fields in fluid flow problems defined in highly heterogeneous porous media, particularly in the presence of boundary layers.
| References | Barrenechea, G. R., & Valentin, F. (2002). An unusual stabilized finite element method for a generalized Stokes problem. Numerische Mathematik, 92(4), 653-677. Pal, M. (2012). A unified approach to simulation and upscaling of single‐phase flow through vuggy carbonates. International journal for numerical methods in fluids, 69(6), 1096-1123. |
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| Country | Brazil |
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