arXiv subjects

Gonzalo Rivera

Publications and source records attributed to Gonzalo Rivera.

At least 19 recordsLinked to original sources

arXiv2026

An augmented mixed finite element method for the Biot problem with nonlinear permeability

In two and three dimensions, we analyze a mixed formulation of the nonlinear Biot equations in terms of the poroelastic stress tensor, the displacement, the pressure, and the rotation tensor. To establish the well-posedness of the problem, we introduce suitable stabilization terms, leading to an augmented mixed formulation in which the associated parameters are chosen appropriately. Owing to the nonlinearity of the problem, we employ a fixed-point strategy to prove the existence and uniqueness of solutions. We propose a finite element scheme for which we establish existence and uniqueness of a discrete solution, together with a priori error estimates. Additionally, we design and analyze an a posteriori error estimator of the residual type which results to be reliable and efficient. Finally, we present a series of numerical experiments in two and three dimensions in order to assess the performance of the proposed method.

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arXiv2026

A locking-free mixed virtual element discretization for the two dimensional elasticity eigenvalue problem

In this paper, we propose and analyze a mixed virtual element method for the approximation of the eigenvalues and eigenfunctions of the two-dimensional elasticity eigenvalue problem. Under standard assumptions on polygonal meshes, we prove the convergence of the discrete solution operator to its continuous counterpart as the mesh size tends to zero. Relying on the spectral theory of compact operators, we establish the spectral correctness of the method and derive error estimates for both eigenvalues and eigenfunctions. A series of numerical experiments is presented to support the theoretical analysis. The results confirm the predicted convergence rates and show that the method is locking-free and able to approximate the spectrum accurately, independently of the shape of the polygonal elements in the mesh.

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arXiv2026

A mixed virtual element discretization for the generalized Oseen problem

In this paper we introduce a mixed virtual element method to approximate the solution for the two dimensional generalized Oseen problem. We introduce the pseudostress as an additional unknown, which allows to eliminate the pressure from the system; the pressure can be recovered via a post-process of the pseudostress tensor. We prove existence and uniqueness of the continuous solution via a fixed point argument. Under standard mesh assumptions, we develop a virtual element method to approximate both the tensor and the velocity field, and we show that it is stable. Furthermore, we provide a priori error estimates for the method and validate them through a series of numerical tests using different polygonal meshes.

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arXiv2025

Discontinuous Galerkin approximation for a Stokes-Brinkman-type formulation for the eigenvalue problem in porous media

We introduce a family of discontinuous Galerkin methods to approximate the eigenvalues and eigenfunctions of a Stokes-Brinkman type of problem based in the interior penalty strategy. Under the standard assumptions on the meshes and a suitable norm, we prove the stability of the discrete scheme. Due to the non-conforming nature of the method, we use the well-known non-compact operators theory to derive convergence and error estimates for the method. We present an exhaustive computational analysis where we compute the spectrum with different stabilization parameters with the aim of study its influence when the spectrum is approximated.

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arXiv2025

A Stokes-Brinkman-type formulation for the eigenvalue problem in porous media

In this paper we introduce and analyze, for two and three dimensions, a finite element method to approximate the natural frequencies of a flow system governed by the Stokes-Brinkman equations. Here, the fluid presents the capability of being within a porous media. Taking advantage of the Stokes regularity results for the solution, and considering inf-sup stable families of finite elements, we prove convergence together with a priori and a posteriori error estimates for the eigenvalues and eigenfunctions with the aid of the compact operators theory. We report a series of numerical tests in order to confirm the developed theory.

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arXiv2024

A noncoforming virtual element approximation for the Oseen eigenvalue problem

In this paper we analyze a nonconforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. The spaces under consideration lead to a divergence-free method which is capable to capture properly the divergence at discrete level and the eigenvalues and eigenfunctions. Under the compact theory for operators we prove convergence and error estimates for the method. By employing the theory of compact operators we recover the double order of convergence of the spectrum. Finally, we present numerical tests to assess the performance of the proposed numerical scheme.

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arXiv2024

A Mixed finite element method for the velocity-pseudostress formulation of the Oseen eigenvalue problem

In this paper, we introduce and analyze a mixed formulation for the Oseen eigenvalue problem by introducing the pseudostress tensor as a new unknown, allowing us to eliminate the fluid pressure. The well-posedness of the solution operator is established using a fixed-point argument. For the numerical analysis, we use the tensorial versions of Raviart-Thomas and Brezzi-Douglas-Marini elements to approximate the pseudostress, and piecewise polynomials for the velocity. Convergence and a priori error estimates are derived based on compact operator theory. We present a series of numerical tests in two and three dimensions to confirm the theoretical findings.

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arXiv2024

A virtual element method for a convective Brinkman-Forchheimer problem coupled with a heat equation

We develop a virtual element method to solve a convective Brinkman-Forchheimer problem coupled with a heat equation. This coupled model may allow for thermal diffusion and viscosity as a function of temperature. Under standard discretization assumptions, we prove the well posedness of the proposed numerical scheme. We also derive optimal error estimates under appropriate regularity assumptions for the solution. We conclude with a series of numerical tests performed with different mesh families that complement our theoretical findings.

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arXiv2023

Error analysis for a non-conforming virtual element discretization of the acoustic problem

We introduce non conforming virtual elements to approximate the eigenvalues and eigenfunctions of the two dimensional acoustic vibration problem. We focus our attention on the pressure formulation of the acoustic vibration problem in order to discretize it with a suitable non conforming virtual space for $\mathrm{H}^1$. With the aid of the theory of non-compact operators we prove convergence and spectral correctness of the method. To illustrate the theoretical results, we report numerical tests on different polygonal meshes, in order to show the accuracy of the method on the approximation of the spectrum.

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arXiv2023

Finite Element Analysis of the Oseen eigenvalue problem

We propose and analyze a finite element method for the Oseen eigenvalue problem. This problem is an extension of the Stokes eigenvalue problem, where the presence of the convective term leads to a non-symmetric problem and hence, to complex eigenvalues and eigenfunctions. With the aid of the compact operators theory, we prove that for inf-sup stable finite elements the convergence holds and hence, error estimates for the eigenvalues and eigenfunctions are derived. We also propose an a posteriori error estimator which results to be reliable and efficient. We report a series of numerical tests in two and three dimension in order to assess the performance of the method and the proposed estimator.

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arXiv2023

VEM allowing small edges for the acoustic problem

In this paper we propose and analyze a virtual element method to approximate the natural frequencies of the acoustic eigenvalue problem with polygonal meshes that allow the presence of small edges. With the aid of a suitable seminorm that depends on the stabilization of the small edges method, we prove convergence and error estimates for the eigenfrequencies and eigenfunctions of the problem, supporting our analysis on the compact operators theory. We report some numerical tests that allows us to assess the performance of the method and the accuracy on the approximation.

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arXiv2023

A priori and a posteriori error analysis for a VEM discretization of the convection-diffusion eigenvalue problem

In this paper we propose and analyze a virtual element method for the two dimensional non-symmetric diffusion-convection eigenvalue problem in order to derive a priori and a posteriori error estimates. Under the classic assumptions of the meshes, and with the aid of the classic theory of compact operators, we prove error estimates for the eigenvalues and eigenfunctions. Also, we develop an a posteriori error estimator which, in one hand, results to be reliable and on the other, with standard bubble functions arguments, also results to be efficient. We test our method on domains where the complex eigenfunctions are not sufficiently regular, in order to assess the performance of the estimator that we compare with the uniform refinement given by the a priori analysis

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arXiv2023

A virtual element method for the elasticity problem allowing small edges

In this paper we analyze a virtual element method for the two dimensional elasticity problem allowing small edges. With this approach, the classic assumptions on the geometrical features of the polygonal meshes can be relaxed. In particular, we consider only star-shaped polygons for the meshes. Suitable error estimates are presented, where a rigorous analysis on the influence of the Lamé constants in each estimate is presented. We report numerical tests to assess the performance of the method.

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arXiv2023

A virtual element method for the elasticity spectral problem allowing small edges

In this paper we analyze a virtual element method for the two dimensional elasticity spectral problem allowing small edges. Under this approach, and with the aid of the theory of compact operators, we prove convergence of the proposed VEM and error estimates, where the influence of the Lamé constants is presented. We present a series of numerical tests to assess the performance of the method where we analyze the effects of the Poisson ratio on the computation of the order of convergence, together with the effects of the stabilization term on the arising of spurious eigenvalues.

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arXiv2023

VEM discretization allowing small edges for the reaction-convection-diffusion equation: source and spectral problems

In this paper we analyze a lowest order virtual element method for the load classic reaction-convection-diffusion problem and the convection-diffusion spectral problem, where the assumptions on the polygonal meshes allow to consider small edges for the polygons. Under well defined seminorms depending on a suitable stabilization for this geometrical approach, we derive the well posedness of the numerical scheme and error estimates for the load problem, whereas for the spectral problem we derive convergence and error estimates for the eigenvalues and eigenfunctions. We report numerical tests to assess the performance of the small edges on our numerical method for both problems under consideration.

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arXiv2022

VEM approximation for the Stokes eigenvalue problem: a priori and a posteriori error analysis

The present paper proposes an inf-sup stable divergence free virtual element method and associated a priori, and a posteriori error analysis to approximate the eigenvalues and eigenfunctions of the Stokes spectral problem in one shot. For the a priori analysis, we take advantage of the compactness of the solution operator to prove convergence of the eigenfunctions and double order convergence of eigenvalues. Additionally we also propose an a posteriori estimator of residual type, which we prove is reliable and efficient, in order to perform adaptive refinements that allow to recover the optimal order of convergence for non smooth eigenfunctions. A set of representative numerical examples investigates such theoretical results.

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arXiv2022

Finite element analysis for the Navier-Lamé eigenvalue problem

The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so called Navier-Lamé system is considered. Such a system introduces the displacement, rotation and pressure of some linear and elastic structure. The analysis of the spectral problem is based in the compact operators theory. A finite element method based in polynomials of degree $k\geq 1$ are considered in order to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimate are presented. An a posteriori error analysis is performed, where the reliability and efficiency of the proposed estimator is proved. We end this contribution reporting a series of numerical tests in order to assess the performance of the proposed numerical method, for the a priori and a posteriori estimates.

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arXiv2022

A posteriori virtual element method for the acoustic vibration problem

In two dimensions, we propose and analyze an a posteriori error estimator for the acoustic spectral problem based on the virtual element method in $\H(÷;Ω)$. Introducing an auxiliary unknown, we use the fact that the primal formulation of the acoustic problem is equivalent to a mixed formulation, in order to prove a superconvergence result, necessary to despise high order terms. Under the virtual element approach, we prove that our local indicator is reliable and globally efficient in the $Ł^2$-norm. We provide numerical results to assess the performance of the proposed error estimator.

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