Error Estimates for a Mixed Virtual Element Approach to the Brinkman Eigenvalue Problem

The Brinkman model provides a mathematical framework for describing viscous fluid flow in porous media, capturing regimes that transition between Darcy and Stokes flows. In this work, we present a Mixed Virtual Element Method (Mixed - VEM) for the numerical approximation of the Brinkman problem, considering both the associated source problem and the corresponding eigenvalue problem. The proposed formulation allows the use of general polygonal meshes, providing
flexibility in the discretization of complex geometries. We discuss the construction of the discrete spaces and the main properties of the method. In particular, we outline the analysis of the source problem and the spectral approximation of the eigenvalue problem, together with error estimates for the relevant variables. Numerical experiments are presented to illustrate the theoretical results and to confirm the performance of the method.