Conforming Virtual Element discretization of the generalized Stokes problem

We develop and analyze a conforming Virtual Element Method on general polygonal meshes for the two-dimensional generalized Stokes equations. This parameter-dependent model provides a continuous transition between the classical Stokes problem and Darcy flow through the viscosity parameter ε. Our discretization employs enhanced virtual element spaces for the velocity, in which the normal trace on each edge is approximated by polynomials of degree one higher than those used for the tangential trace. We prove that the method is inf-sup stable with respect to the parameter ε. We provide the a priori error analysis in two different ϵ-weighted norms. In the first case, the a priori error analysis yields an optimal-order estimate for the velocity in a natural ε-weighted norm and a pressure estimate in the L 2 norm also reveals the ε-dependency of the constants that appear in the priori analysis. In the second case, the weighted velocity estimate is uniform with respect to ε, which ensures robust approximation across the full Stokes–Darcy transition regime. We assess the performance of the method on several numerical tests using different families of polygonal meshes, including triangular, quadrilateral, Voronoi, and non-convex meshes.