Preprint 2026-12
Sergio Caucao, Gabriel N. Gatica, Luis F. Gatica, Cristian Inzunza:
A priori and a posteriori error analysis of a mixed FEM for stationary convective Brinkman-Forchheimer flows with variable porosity
Abstract:
We propose and analyze a mixed finite element method for stationary convective Brinkman--Forchheimer flows with variable porosity in $\R^d$, $d \in \{2,3\}$. While the primary unknowns are the velocity and pressure, our approach is based on introducing a nonlinear pseudostress as an additional variable, which allows us to eliminate the pressure from the formulation. Nevertheless, the latter, along with other physically relevant quantities such as the velocity gradient, vorticity, and stress tensor, can be accurately recovered through postprocessing formulas that depend mainly on the pseudostress and velocity. This capability constitutes one of the most distinctive features of the proposed strategy. Owing to the convective and Forchheimer terms, the velocity must be sought in a smaller space than in the standard setting, which naturally leads to a Banach space framework. The resulting formulation exhibits a perturbed saddle-point structure and can be equivalently recast as a fixed-point equation. Under suitable small-data assumptions, the unique solvability of the continuous and discrete problems is established by combining the Banach fixed-point theorem with the Babu\v{s}ka--Brezzi theory in Banach spaces for perturbed saddle-point problems and the Banach--Ne\v{c}as--Babu\v{s}ka theorem. The finite element method employs Raviart--Thomas spaces of order $k\ge 0$ and spaces of discontinuous piecewise polynomials of degree $k$ for the nonlinear pseudostress tensor and the velocity, respectively. Stability, convergence, and \textit{a priori} error estimates are derived. We also derive a reliable and efficient residual-based \textit{a posteriori} error estimator on general polygonal and polyhedral domains. Finally, several numerical examples illustrate the performance of the method, confirm the theoretical convergence rates, validate the estimator, and demonstrate the behavior of the associated adaptive algorithm, including the case of flow through a two-dimensional nonconvex channel with localized low-permeability regions.
