Dr. Caucao-Paillán, Sergio

2024, Dr. Caucao-Paillán, Sergio, Esparza, Johann

We propose and analyse an augmented mixed finite element method for the pseudo stress–velocity formulation of the stationary convective Brinkman–Forchheimer problem inRd, d∈ {2,3}. Since the convective and Forchheimer terms forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram theorem, allow to prove the unique solvability of the continuous problem. The finite element discretization involves Raviart–Thomas spaces of order k≥0 for the pseudostress tensor and continuous piecewise polynomials of degree ≤k+1 for the velocity. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. In addition, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the form involved, a suitable assumption on the data, a stable Helmholtz decomposition, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, some numerical examples illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the properties of the estimators, and showing the behaviour of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with fracture networks is considered.

2023, Caucao-Paillán, Sergio, Gatica, Gabriel, Ortega, Juan

In this paper we consider a Banach spaces-based fully-mixed variational formulation that has been recently proposed for the coupling of the stationary Brinkman–Forchheimer and double-diffusion equations, and develop the first reliable and efficient residual-based a posteriori error estimator for the 2D and 3D versions of the associated mixed finite element scheme. For the reliability analysis, and due to the nonlinear nature of the problem, we employ the strong monotonicity of the operator involving the Forchheimer term, in addition to inf-sup conditions of some of the resulting bilinear forms, along with a stable Helmholtz decomposition in nonstandard Banach spaces, which, in turn, having been recently derived, constitutes another distinctive feature of the paper, and local approximation properties of the Raviart–Thomas and Clément interpolants. On the other hand, inverse inequalities, the localization technique through bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with an irregular channel networks is considered.

2022, Dr. Caucao-Paillán, Sergio, Gatica, Gabriel, Oyarzúa, Ricardo, Zúñiga, Paulo

In this paper we consider a partially augmented fully-mixed variational formulation that has been recently proposed for the coupling of the stationary Brinkman–Forchheimer and double-diffusion equations, and develop an a posteriori error analysis for the 2D and 3D versions of the associated mixed finite element scheme. Indeed, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary (convex or non-convex) polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity and inf-sup condition of the forms involved, a suitable assumption on the data, stable Helmholtz decompositions in Hilbert and Banach frameworks, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimators and illustrating the performance of the associated adaptive algorithms, are reported. In particular, the case of flow through a 3D porous media with channel networks is considered.