arXiv subjects

Gilberto Campaña

Publications and source records attributed to Gilberto Campaña.

5 recordsLinked to original sources

arXiv2025

Bilinear optimal control for the Stokes-Brinkman equations: a priori and a posteriori error analyses

We analyze a bilinear optimal control problem for the Stokes--Brinkman equations: the control variable enters the state equations as a coefficient. In two- and three-dimensional Lipschitz domains, we perform a complete continuous analysis that includes the existence of solutions and first- and second-order optimality conditions. We also develop two finite element methods that differ fundamentally in whether the admissible control set is discretized or not. For each of the proposed methods, we perform a convergence analysis and derive a priori error estimates; the latter under the assumption that the domain is convex. Finally, assuming that the domain is Lipschitz, we develop an a posteriori error estimator for each discretization scheme, obtain a global reliability bound, and investigate local efficiency estimates.

math.NA↗

arXiv2024

Finite element approximation for a convective Brinkman--Forchheimer problem coupled with a heat equation

We investigate a convective Brinkman--Forchheimer problem coupled with a heat equation. The investigated model considers thermal diffusion and viscosity depending on the temperature. We prove the existence of a solution without restriction on the data and uniqueness when the solution is slightly smoother and the data is suitably restricted. We also propose a finite element discretization scheme for the considered model and derive convergence results and a priori error estimates. Finally, we illustrate the theory with numerical examples.

math.NA↗

arXiv2024

Numerical discretizations of a convective Brinkman-Forchheimer model under singular forcing

In two-dimensional Lipschitz domains, we analyze a Brinkman--Darcy--Forchheimer problem on the weighted spaces $\mathbf{H}_0^1(ω,Ω) \times L^2(ω,Ω)/\mathbb{R}$, where $ω$ belongs to the Muckenhoupt class $A_2$. Under a suitable smallness assumption, we prove the existence and uniqueness of a solution. We propose a finite element method and obtain a quasi-best approximation result in the energy norm \emph{à la Céa} under the assumption that $Ω$ is convex. We also develop an a posteriori error estimator and study its reliability and efficiency properties. Finally, we develop an adaptive method that yields optimal experimental convergence rates for the numerical examples we perform.

math.NA↗

arXiv2023

The linear elasticity system under singular forces

We study the linear elasticity system subject to singular forces. We show existence and uniqueness of solutions in two frameworks: weighted Sobolev spaces, where the weight belongs to the Muckenhoupt class $A_2$; and standard Sobolev spaces where the integrability index is less than $d/(d-1)$; $d$ is the spatial dimension. We propose a standard finite element scheme and provide optimal error estimates in the $\mathbf{L}^2$--norm. By proving well posedness, we clarify some issues concerning the study of generalized mixed problems in Banach spaces.

math.NA↗

arXiv2023

Numerical discretization of a Darcy-Forchheimer problem coupled with a singular heat equation

In Lipschitz domains, we study a Darcy-Forchheimer problem coupled with a singular heat equation by a nonlinear forcing term depending on the temperature. By singular we mean that the heat source corresponds to a Dirac measure. We establish the existence of solutions for a model that allows a diffusion coefficient in the heat equation depending on the temperature. For such a model, we also propose a finite element discretization scheme and provide an a priori convergence analysis. In the case that the aforementioned diffusion coefficient is constant, we devise an a posteriori error estimator and investigate reliability and efficiency properties. We conclude by devising an adaptive loop based on the proposed error estimator and presenting numerical experiments.

math.NA↗