Preprint 2026-03
Sergio Caucao, Gabriel N. Gatica, Adrian Suarez, Ivan Yotov:
A skew-symmetry-based mixed formulation for an Oseen-type Kelvin--Voigt--Brinkman--Forchheimer model
Abstract:
We propose and analyze a new mixed formulation for an Oseen linearization of the Kelvin--Voigt--Brinkman--Forchheimer equations, which model viscoelastic flows at higher velocities in highly porous media. Besides the velocity field, our approach introduces the velocity gradient and the viscoelastic pseudostress tensors as auxiliary unknowns, thereby allowing the pressure to be eliminated from the system while still being recoverable through a simple postprocess. This leads to a three-field mixed variational formulation within a Banach space framework, where the aforementioned variables constitute the main unknowns, exploiting the skew-symmetric structure of one of the operators involved. Existence and uniqueness of a solution to the weak formulation are established, and the corresponding stability bounds are derived by employing classical results on nonlinear monotone operators. A semidiscrete continuous-in-time approximation is then introduced, based on Raviart--Thomas finite elements of degree $k \ge 0$ for the viscoelastic pseudostress tensor and discontinuous piecewise polynomials of degree $k$ for the velocity and velocity gradient fields. Furthermore, by applying a backward Euler time discretization, a fully discrete finite element scheme is obtained. Well-posedness is established, and stability bounds together with the corresponding {\it a priori} error estimates are derived for both schemes. Several numerical experiments involving both manufactured and non-manufactured solutions are presented, confirming the theoretical convergence rates and illustrating the capability of the proposed method to handle challenging geometries with strong contrasts in physical parameters such as permeability and elasticity.
