Publications | Jesus Vellojin

Journal Articles

2025

Nonconforming virtual element method for the Schrödinger eigenvalue problem

Dibyendu Adak , Gianmarco Manzini , and Jesus Vellojin

Computers & Mathematics with Applications, 2025

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This study presents an in-depth analysis of the nonconforming virtual element method (VEM) as a novel approach for approximating the eigenvalues of the Schrödinger equation. Central to the strategy is deploying the L2 projection operator to discretize potential terms within the model problem. Through compact operator theory, we rigorously establish the methodology’s capability to achieve double-order convergence rates for the eigenvalue spectrum. Addressing the challenge posed by the nonconformity of the discrete space, we redefine the solution operator on a weaker space, which aligns with the Babuška-Osborn compactness framework. A comprehensive set of numerical experiments confirms the theoretical findings, showing the approximation qualities and computational efficiency of the method. A series of potential functions are used to illustrate the various challenges behind the choice of a potential for the simulation of the Schrödinger eigenvalue problem. These results confirm the potential of the nonconforming VEM as a robust and accurate tool for quantum mechanical eigenvalue problems.
Finite element analysis for the Navier-Lamé eigenvalue problem

Felipe Lepe , Gonzalo Rivera , and Jesus Vellojin

Applied Numerical Mathematics, 2025

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The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so-called Navier-Lamé system is considered. This system incorporates the displacement, rotation, and pressure of a linear elastic structure. The analysis of the spectral problem is based on the compact operator theory. A finite element method using polynomials of degree k ≥1 is employed to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimates are presented. An a posteriori error analysis is also performed, where the reliability and efficiency of the proposed estimator are proven. We conclude this contribution by reporting a series of numerical tests to assess the performance of the proposed numerical method for both a priori and a posteriori estimates.

2024

Discontinuous Galerkin methods for the acoustic vibration problem

Felipe Lepe , David Mora , and Jesus Vellojin

Journal of Computational and Applied Mathematics, 2024

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In two and three dimension we analyze discontinuous Galerkin methods (DG) for the acoustic vibration problem. Through all our study we consider an inviscid fluid, leading to a linear eigenvalue problem. The acoustic problem is written, in first place, in terms of the displacement. Under the approach of the non-compact operators theory, we prove convergence and error estimates for the method when the displacement formulation is considered. We analyze the influence of the stabilization parameter on the computation of the spectrum, where spurious eigenmodes arise when this parameter is not correctly chosen. Alternatively we present the formulation depending only on the pressure, comparing the performance of the DG methods with the pure displacement formulation. Computationally, we study the influence of the stabilization parameter on the arising of spurious eigenvalues when the spectrum is computed. Also, we report tests in two and three dimensions where convergence rates are reported, together with a comparison between the displacement and pressure formulations for the proposed DG methods.
A finite element model for concentration polarization and osmotic effects in a membrane channel

Nicolás Carro , David Mora , and Jesus Vellojin

International Journal for Numerical Methods in Fluids, 2024

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Abstract In this article, we study a mathematical model that represents the concentration polarization and osmosis effects in a reverse osmosis cross-flow channel with dense membranes at some of its boundaries. The fluid is modeled using the Navier–Stokes equations and the solution-diffusion is used to impose the momentum balance on the membrane. The scheme consist of a conforming finite element method with the velocity–pressure formulation for the Navier–Stokes equations, together with a primal scheme for the convection–diffusion equations. The Nitsche’s method is used to impose the permeability condition across the membrane. Several numerical experiments are performed to show the robustness of the method. The resulting model accurately replicates the analytical models and predicts similar results to previous works. It is found that the submerged configuration has the highest permeate production, but also has the greatest pressure loss of all three configurations studied.
Finite element analysis of the Oseen eigenvalue problem

Felipe Lepe , Gonzalo Rivera , and Jesus Vellojin

Computer Methods in Applied Mechanics and Engineering, 2024

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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.
A mixed parameter formulation with applications to linear viscoelastic slender structures

Hernández, Erwin , Lepe, Felipe , and Vellojin, Jesus

ESAIM: M2AN, 2024

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We present the analysis of an abstract parameter-dependent mixed variational formulation based on Volterra integrals of second kind. Adapting the classic mixed theory in the Volterra equations setting, we prove the well posedness of the resulting system. Stability and error estimates are derived, where all the estimates are independent of the perturbation parameter. We provide applications of the developed analysis for a viscoelastic Timoshenko beam and report numerical tests for this problem. We also comment, numerically, the performance of a viscoelastic Reissner-Mindlin plate.

2023

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

Felipe Lepe , and Jesus Vellojin

Calcolo, 2023

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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.
Error estimates for a vorticity-based velocity–stress formulation of the Stokes eigenvalue problem

Felipe Lepe , Gonzalo Rivera , and Jesus Vellojin

Journal of Computational and Applied Mathematics, 2023

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The aim of this paper is to analyze a mixed formulation for the two dimensional Stokes eigenvalue problem where the unknowns are the stress and the velocity, whereas the pressure can be recovered with a simple postprocess of the stress. The stress tensor is written in terms of the vorticity of the fluid, leading to an alternative mixed formulation that incorporates this physical feature. We propose a mixed numerical method where the stress is approximated with suitable Nédelec finite elements, whereas the velocity is approximated with piecewise polynomials of degree k≥0. With the aid of the compact operators theory we derive convergence of the method and spectral correctness. Moreover, we propose a reliable and efficient a posteriori error estimator for our spectral problem in order to provide an adaptive strategy to achieve the optimal order of convergence for non sufficient smooth eigenfunctions. We report numerical tests where the spectrum is computed, together with a computational analysis for the proposed estimator. In addition, we use the corresponding error estimator to drive an adaptive scheme, and we report the results of a numerical test, that allow us to assess the performance of this approach.

2022

A Posteriori Analysis for a Mixed FEM Discretization of the Linear Elasticity Spectral Problem

Felipe Lepe , Gonzalo Rivera , and Jesus Vellojin

Journal of Scientific Computing, 2022

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In this paper we analyze a posteriori error estimates for a mixed formulation of the linear elasticity eigenvalue problem. A posteriori estimators for the nearly and perfectly compressible elasticity spectral problems are proposed. With a post-process argument, we are able to prove reliability and efficiency for the proposed estimators. The numerical method is based in Raviart-Thomas elements to approximate the pseudostress and piecewise polynomials for the displacement. We illustrate our results with numerical tests in two and three dimensions.
Mixed Methods for the Velocity-Pressure-Pseudostress Formulation of the Stokes Eigenvalue Problem

Felipe Lepe , Gonzalo Rivera , and Jesus Vellojin

SIAM Journal on Scientific Computing, 2022

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In two and three dimensions, we analyze mixed finite element methods for a velocity-pressure-pseudostress formulation of the Stokes eigenvalue problem. The methods consist of two schemes: the velocity and pressure are approximated with piecewise polynomial, whereas for the pseudostress we consider two classic families of finite elements for {H(\div) spaces: the Raviart–Thomas and the Brezzi–Douglas–Marini elements. With the aid of the classic spectral theory for compact operators, we prove that our method does not introduce spurious modes. Also, we obtain convergence and error estimates for the proposed methods. We report numerical results to compare the accuracy and robustness between both numerical schemes.

2021

A locking-free finite element formulation for a non-uniform linear viscoelastic Timoshenko beam

Erwin Hernández , and Jesus Vellojin

Computers & Mathematics with Applications, 2021

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In this paper we use a modified constitutive law in a hereditary integral form to analyze the response of an isotropic non-uniform linear viscoelastic Timoshenko beam. A mixed method framework is used to provide stability and semi-discrete error estimates that do not deteriorate as the thickness parameter becomes small. Numerical experiments for both quasi-static and transient cases are presented.

2020

Modelling of thin viscoelastic shell structures under Reissner–Mindlin kinematic assumption

E. Hernandez , C. Naranjo , and J. Vellojin

Applied Mathematical Modelling, 2020

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In this paper, we study thin viscoelastic shell structures using a constitutive equation in hereditary integral form. An alternative mathematical formulations for several viscoelastic shell structures under the Reissner–Mindlin kinematical assumptions are obtained. The resulting equations are written as a Volterra equation of the second kind to allow further mathematical analysis. A locking-free finite element formulation, with selective reduced integration is used to approximate the equation. To perform numerical experiments we consider several situations suffering from locking in both cases dynamic and quasi-static. We show the good behavior of the model compared with other models from the literature.

Preprints

2024

Divergence conforming finite element methods for flow-transport coupling with osmotic effects

Arbaz Khan , David Mora , Ricardo Ruíz-Baier , and Jesus Vellojin

2024

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We propose a model for the coupling of flow and transport equations with porous membrane-type conditions on part of the boundary. The governing equations consist of the incompressible Navier–Stokes equations coupled with an advection-diffusion equation, and we employ a Lagrange multiplier to enforce the coupling between penetration velocity and transport on the membrane, while mixed boundary conditions are considered in the remainder of the boundary. We show existence and uniqueness of the continuous problem using a fixed-point argument. Next, an H(div)-conforming finite element formulation is proposed, and we address its a priori error analysis. The method uses an upwind approach that provides stability in the convection-dominated regime. We showcase a set of numerical examples validating the theory and illustrating the use of the new methods in the simulation of reverse osmosis processes.
A Mixed finite element method for the velocity-pseudostress formulation of the Oseen eigenvalue problem

Felipe Lepe , Gonzalo Rivera , and Jesus Vellojin

2024

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

Arbaz Khan , Felipe Lepe , and Jesus Vellojin

2024

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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 Lamé 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 Lamé constants. Additionally, we conduct an a posteriori analysis and propose a reliable and efficient estimator. The theoretical findings are supported by numerical experiments.
Nitsche stabilized Virtual element approximations for a Brinkman problem with mixed boundary conditions

David Mora , Jesus Vellojin, and Nitesh Verma

2024

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In this paper, we formulate, analyse and implement the discrete formulation of the Brinkman problem with mixed boundary conditions, including slip boundary condition, using the Nitsche’s technique for virtual element methods. The divergence conforming virtual element spaces for the velocity function and piecewise polynomials for pressure are approached for the discrete scheme. We derive a robust stability analysis of the Nitsche stabilized discrete scheme for this model problem. We establish an optimal and vigorous a priori error estimates of the discrete scheme with constants independent of the viscosity. Moreover, a set of numerical tests demonstrates the robustness with respect to the physical parameters and verifies the derived convergence results.

2023

Finite element analysis of the nearly incompressible linear elasticity eigenvalue problem with variable coefficients

Arbaz Khan , Felipe Lepe , David Mora , and Jesus Vellojin

2023

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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 Babuška–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 Lamé coefficient λ. 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.