arXiv subjects

Ruben Caraballo

Publications and source records attributed to Ruben Caraballo.

4 recordsLinked to original sources

arXiv2025

Discontinuous in time Virtual Element method for Darcy equations coupled with Multi Species Transport with First Order Reaction Network

We study transport phenomena involving chemically reactive species, modeled by advection--diffusion--reaction systems coupled with flow fields governed by Darcy's law. Both the velocity field and the species concentrations are discretized using the Virtual Element Method, while time integration is performed through a discontinuous Galerkin scheme. This work represents a preliminary study, in which we introduce some simplifications of the full model. In particular, we assume a concentration-independent viscosity in the Darcy problem, constant diffusion tensors in the advection--diffusion--reaction systems, and first-order reaction networks with liquid-phase degradation. We derive an abstract error estimate by means of a technique that combines Gauss--Radau interpolation with numerical integration. The theoretical results are supported by numerical experiments that exhibit arbitrary-order accuracy in both space and time.

math.NA↗

arXiv2023

Robust finite element methods and solvers for the Biot--Brinkman equations in vorticity form

In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lam\'e parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.

math.NA↗

arXiv2022

On augmented finite element formulation for the Navier--Stokes equations with vorticity and variable viscosity

We propose and analyse an augmented mixed finite element method for the Navier--Stokes equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and no-slip boundary conditions. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition. The theoretical and practical implications of using augmentation is discussed in detail. In addition, we use fixed--point strategies to show the existence and uniqueness of continuous and discrete solutions under the assumption of sufficiently small data. The method is constructed using any compatible finite element pair for velocity and pressure as dictated by Stokes inf-sup stability, while for vorticity any generic discrete space (of arbitrary order) can be used. We establish optimal a priori error estimates. Finally, we provide a set of numerical tests in 2D and 3D illustrating the behaviour of the scheme as well as verifying the theoretical convergence rates.

math.NA↗

arXiv2021

Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity

We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babu\vska-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation.

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