arXiv subjects

Iván Velásquez

Publications and source records attributed to Iván Velásquez.

7 recordsLinked to original sources

arXiv2026

A posteriori analysis of a virtual element approach on polytopal meshes for the buckling eigenvalue problem

We introduce a novel residual-based a posteriori error estimator for the conforming $C^1$ Virtual Element Method (VEM) applied to the buckling eigenvalue problem, incorporating nonlinear plane stress effects in both two and three dimensions. The estimator is fully computable on general polyhedral meshes and implemented within the open-source \texttt{vem++} library. Its reliability is rigorously justified via bounds on the residual equation using polynomial projections, stabilisation contributions, and interpolation estimates, while efficiency is ensured through the use of bubble function arguments. Comprehensive numerical experiments in 2D and 3D illustrate the estimator's optimal accuracy and robustness, highlighting its potential for predictive analysis of complex plate structures.

math.NA↗

arXiv2025

A posteriori error estimates for a $C^1$ virtual element method applied to the thin plate vibration problem

We propose and analyse residual-based a posteriori error estimates for the virtual element discretisation applied to the thin plate vibration problem in both two and three dimensions. Our approach involves a conforming $C^1$ discrete formulation suitable for meshes composed of polygons and polyhedra. The reliability and efficiency of the error estimator are established through a dimension-independent proof. Finally, several numerical experiments are reported to demonstrate the optimal performance of the method in 2D and 3D.

math.NA↗

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.

math.NA↗

arXiv2020

A virtual element method for the Steklov eigenvalue problem allowing small edges

The aim of this paper is to analyze the influence of small edges in the computation of the spectrum of the Steklov eigenvalue problem by a lowest order virtual element method. Under weaker assumptions on the polygonal meshes, which can permit arbitrarily small edges with respect to the element diameter, we show that the scheme provides a correct approximation of the spectrum and prove optimal error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we report some numerical tests supporting the theoretical results.

math.NA↗

arXiv2019

Virtual Element for the Buckling Problem of Kirchhoff-Love plates

In this paper, we develop a virtual element method (VEM) of high order to solve the fourth order plate buckling eigenvalue problem on polygonal meshes. We write a variational formulation based on the Kirchhoff-Love model depending on the transverse displacement of the plate. We propose a $C^1$ conforming virtual element discretization of arbitrary order $k\ge2$ and we use the so-called Babuska--Osborn abstract spectral approximation theory to show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the buckling modes (eigenfunctions) and a double order for the buckling coefficients (eigenvalues). Finally, we report some numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes.

math.NA↗

arXiv2018

A Virtual Element Method for the Transmission Eigenvalue Problem

In this paper, we analyze a virtual element method (VEM) for solving a non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a $C^1$-conforming discretization by means of the VEM. We use the classical approximation theory for compact non-selfadjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes.

math.NA↗

arXiv2017

A virtual element method for the vibration problem of Kirchhoff plates

The aim of this paper is to develop a virtual element method (VEM) for the vibration problem of thin plates on polygonal meshes. We consider a variational formulation relying only on the transverse displacement of the plate and propose an $H^2(Ω)$ conforming discretization by means of the VEM which is simple in terms of degrees of freedom and coding aspects. Under standard assumptions on the computational domain, we establish that the resulting schemeprovides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. The analysis restricts to simply connected polygonal clamped plates, not necessarily convex. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes. Additional examples of cases not covered by our theory are also presented.

math.NA↗