Dr. Caucao-Paillán, Sergio

2022, Dr. Caucao-Paillán, Sergio, Oyarzúa, Ricardo, Villa-Fuentes, Segundo, Yotov, Ivan

We propose and analyze a new mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Besides the velocity, our approach introduces the velocity gradient and a pseudostress tensor as further unknowns. As a consequence, we obtain a three-field Banach spaces-based mixed variational formulation, where the aforementioned variables are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation, and derive the corresponding stability bounds, employing classical results on nonlinear monotone operators. We then propose a semidiscrete continuous-in-time approximation on simplicial grids based on the Raviart–Thomas elements of degree k ≥ 0 for the pseudostress tensor and discontinuous piecewise polynomials of degree k for the velocity and the velocity gradient. In addition, by means of the backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove wellposedness and derive the stability bounds for both schemes, and under a quasi-uniformity assumption on the mesh, we establish the corresponding error estimates. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.

2022, Dra. Camaño-Valenzuela, Jessika, Dr. Caucao-Paillán, Sergio, Oyarzúa, Ricardo, Villa-Fuentes, Segundo

In this paper we develop an a posteriori error analysis of a new momentum conservative mixed finite element method recently introduced for the steady-state Navier–Stokes problem in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, and suitable Helmholtz decompositions, we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed finite element scheme on arbitrary (convex or non-convex) polygonal and polyhedral regions. On the other hand, inverse inequalities, the localization technique based on bubble functions, among other tools, are employed to prove the efficiency of the proposed a posteriori error indicator. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

2022, Dr. Caucao-Paillán, Sergio, Oyarzúa, Ricardo, Villa-Fuentes, Segundo

In this paper we complement the study of a new mixed finite element scheme, allowing conservation of momentum and thermal energy, for the Boussinesq model describing natural convection and derive a reliable and efficient residual-based a posteriori error estimator for the corresponding Galerkin scheme in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, suitable Helmholtz decompositions and the local approximation properties of the Clément and Raviart–Thomas operators, we derive the aforementioned a posteriori error estimator on arbitrary (convex or non-convex) polygonal and polyhedral regions. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are employed to prove the local efficiency of the proposed a posteriori error estimator. Finally, to illustrate the performance of the adaptive algorithm based on the proposed a posteriori error indicator and to corroborate the theoretical results, we provide some numerical examples.

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.