arXiv subjects

Felipe Lepe

Publications and source records attributed to Felipe Lepe.

At least 19 recordsLinked to original sources

arXiv2026

A virtual element method for Darcy's problem coupled with a heat equation

In two dimensions, we develop a virtual element method to solve the Darcy's problem coupled with a nonlinear heat equation. This coupled model may allow for thermal diffusion and viscosity as a function of temperature. Under standard discretization assumptions and appropriate assumptions on the data, we prove the well posedness of the proposed numerical scheme. We also derive optimal error estimates under suitable regularity assumptions for the solution and appropriate assumptions on the data. We conclude with a series of numerical tests performed on different families of meshes that complement the theoretical findings.

math.NA↗

arXiv2026

A locking free mixed FEM based on a pure pseudostress based formulation for the elasticity eigenproblem

We analyze a novel locking-free mixed formulation for the elasticity eigenvalue problem in both two and three dimensions, expressed exclusively in terms of the pseudostress tensor. An important feature of this formulation is that it does not require the enforcement of symmetry, either in a weak or strong sense. The displacement of the structure is recovered via a postprocess of the computed pseudostress. We introduce a mixed finite element method based in the tensorial version of the standard families of finite elements to discretize the space $\boldsymbol{\mathcal{H}}(\bdiv)$. We prove convergence and a priori error estimates under the theory of non-compact operators. Additionally, we perform an a posteriori error analysis for the problem, proving reliability and efficiency of the proposed indicator. We validate our theoretical results with numerical tests on different geometrical and physical configurations.

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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.

math.NA↗

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.

math.NA↗

arXiv2025

Finite element analysis for a Herrmann pressure formulation of the elastoacoustic problem with variable coefficients

In two and three dimensions, this study is focused on the numerical analysis of an eigenproblem associated with a fluid-structure model for sloshing and elasto-acoustic vibration. We use a displacement-Herrmann pressure formulation for the solid, while for the fluid, a pure displacement formulation is considered. Under this approach we propose a non conforming locking-free method based on classic finite elements to approximate the natural frequencies (of the eigenmodes) of the coupled system. Employing the theory for non-compact operators we prove convergence and error estimates. Also we propose an a posteriori error estimator for this coupled problem which is shown to be efficient and reliable. All the presented theory is contrasted with a set of numerical tests in 2D and 3D.

math.NA↗

arXiv2025

Finite element analysis of an eigenvalue problem arising from neutron transport

In two and three dimensions, we analyze a finite element method to approximate the solutions of an eigenvalue problem arising from neutron transport. We derive the eigenvalue problem of interest, which results to be non-symmetric. Under a standard finite element approximation based on piecewise polynomials of degree $k \geq 1$, and under the framework of the compact operators theory, we prove convergence and error estimates of the proposed method. We report a series of numerical tests in order confirm the theoretical results.

math.NA↗

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.

math.NA↗

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.

math.NA↗

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.

math.NA↗

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.

math.NA↗

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.

math.NA↗

arXiv2024

A Conforming virtual element approximation for the Oseen eigenvalue problem

In this paper we analyze a conforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. We consider the classic velocity-pressure formulation which allows us to consider the divergence-conforming virtual element spaces employed for the Stokes equations. Under standard assumptions on the meshes we derive a priori error estimates for the proposed method with the aid of the compact operators theory. We report some numerical tests to confirm the theoretical results.

math.NA↗

arXiv2024

Error estimates for a bilinear optimal control problem of Maxwell's equations

We consider a control-constrained optimal control problem subject to time-harmonic Maxwell's equations; the control variable belongs to a finite-dimensional set and enters the state equation as a coefficient. We derive existence of optimal solutions, and analyze first- and second-order optimality conditions. We devise an approximation scheme based on the lowest order N\'ed\'elec finite elements to approximate optimal solutions. We analyze convergence properties of the proposed scheme and prove a priori error estimates. We also design an a posteriori error estimator that can be decomposed as the sum two contributions related to the discretization of the state and adjoint equations, and prove that the devised error estimator is reliable and locally efficient. We perform numerical tests in order to assess the performance of the devised discretization strategy and the a posteriori error estimator.

math.NA↗

arXiv2024

Interior penalty discontinuous Galerkin methods for the nearly incompressible elasticity eigenvalue problem with heterogeneous media

This paper studies the family of interior penalty discontinuous Galerkin methods for solving the Herrmann formulation of the linear elasticity eigenvalue problem in heterogeneous media. By employing a weighted Lam\'e coefficient norm within the framework of non-compact operators theory, we prove convergence of both continuous and discrete eigenvalue problems as the mesh size approaches zero, independently of the Lam\'e constants. Additionally, we conduct an a posteriori analysis and propose a reliable and efficient estimator. The theoretical findings are supported by numerical experiments.

math.NA↗

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 nearly incompressible linear elasticity eigenvalue problem with variable coefficients

In this paper we present a mathematical and numerical analysis of an eigenvalue problem associated to the elasticity-Stokes equations stated in two and three dimensions. Both problems are related through the Herrmann pressure. Employing the Babu\v ska--Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are well-posed. In particular, the finite element method is based on general inf-sup stables pairs for the Stokes system, such that, Taylor--Hood finite elements. By using a general approximation theory for compact operators, we obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Under mild assumptions, we have that these estimates hold with constants independent of the Lam\'e coefficient $\lambda$. In addition, we carry out the reliability and efficiency analysis of a residual-based a posteriori error estimator for the spectral problem. We report a series of numerical tests in order to assess the performance of the method and its behavior when the nearly incompressible case of elasticity is considered.

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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.

math.NA↗

arXiv2023

A posteriori analysis for a mixed formulation of the Stokes spectral problem

In two and three dimensions, we design and analyze a posteriori error estimators for the mixed Stokes eigenvalue problem. The unknowns on this mixed formulation are the pseudotress, velocity and pressure. With a lowest order mixed finite element scheme, together with a postprocressing technique, we prove that the proposed estimator is reliable and efficient. We illustrate the results with several numerical tests in two and three dimensions in order to assess the performance of the estimator.

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