A Virtual Element Method for the Steklov Eigenvalue Problem Allowing Small Edges

Lepe, Felipe; Mora, David; Rivera, Gonzalo; Velásquez, Iván

doi:10.1007/s10915-021-01555-3

Cited by 17 publications

(7 citation statements)

References 44 publications

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“…This stabilization is the one considered in, for instance, [8] for the Laplace source problem and [17] for the Steklov eigenvalue problem. With this stabilization, we perform the computation of the convergence orders for the elasticity problem for different values of the Poisson ratio and hence, different values of the Lamé coefficient λ S .…”

Section: • Tmentioning

confidence: 99%

“…The aim now is to analyze the performance of the VEM with small edges in other contexts, in fact, this research is in ongoing progress. For instance, in [17] an application of this new approach in eigenvalue problems has shown the accuracy of the approximation of the spectrum for the Steklov eigenvalue problem, in [23] for elliptic interface problems, [14] for three dimensional problems considering polytopal meshes, etc.…”

Section: Introductionmentioning

confidence: 99%

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A virtual element method for the elasticity problem allowing small edges

Amigo¹,

Lepe²,

Rivera³

2022

Preprint

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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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Section: • Tmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A virtual element method for the elasticity problem allowing small edges

Amigo¹,

Lepe²,

Rivera³

2022

Preprint

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“…This is an ongoing subject of research, and the available results are for VEM spaces to discretize H 1 . Also, recently on [3,20,27,32] is possible to find applications of the small edges approach.…”

Section: Introductionmentioning

confidence: 99%

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

Lepe¹,

Rivera²

2023

Preprint

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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 fo the eigenvalues and eigenfunctions. We report numerical tests to asses the performance of the small edges on our numerical method for both problems under consideration.

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“…The VEM introduced in [12] as a generalization of FEM which is characterized by the capability of dealing with very general polygonal/polyhedral meshes, including hanging nodes and nonconvex elements (see [10,7,13,15,19,20,26,27,30,32,33,37] and refereneces therein). The VEM also permits to easily implement highly regular conforming discrete spaces [18,22] which make the method very feasible to solve various fourth-order problems [8,35,14,34,36].…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

Adak¹,

Mora²,

Natarajan³

2022

Preprint

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In this article, we have considered a nonlinear nonlocal time dependent fourth order equation demonstrating the deformation of a thin and narrow rectangular plate. We propose C 1 conforming virtual element method (VEM) of arbitrary order, k ≥ 2, to approximate the model problem numerically. We employ VEM to discretize the space variable and fully implicit scheme for temporal variable. Well-posedness of the fully discrete scheme is proved under certain conditions on the physical parameters, and we derive optimal order of convergence in both space and time variable. Finally, numerical experiments are presented to illustrate the behaviour of the proposed numerical scheme.

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A Virtual Element Method for the Steklov Eigenvalue Problem Allowing Small Edges

Cited by 17 publications

References 44 publications

A virtual element method for the elasticity problem allowing small edges

A virtual element method for the elasticity problem allowing small edges

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

Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation

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