Virtual element methods for general linear elliptic interface problems on polygonal meshes with small edges

Tushar, Jai; Kumar, Anil; Kumar, Sarvesh

doi:10.1016/j.camwa.2022.07.016

Cited by 15 publications

(3 citation statements)

References 37 publications

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“…In [17], the authors propose an interface-fitted shape regular polytopal mesh generator and virtual element methods for elliptic interface problems. Tushar et al [36] extend the analysis for the finite element method in [18] to VEM for the two dimensional elliptic interface problem and obtain nearly optimal error estimates under realistic assumptions. All the existing VEMs for the interface problem are conforming.…”

Section: Introductionmentioning

confidence: 99%

A Nonconforming Virtual Element Method for the Elliptic Interface Problem

Wang,

Zheng,

null

et al. 2024

EAJAM

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In this paper, we propose a nonconforming virtual element method for the elliptic interface problem based on an unfitted polygonal mesh. On interface elements, the intersecting points of the interface and the edges of elements are considered as additional nodes of the mesh, and then we present a virtual element space satisfying the interface conditions. On non-interface elements, we use the usual nonconforming virtual element. By employing a computable operator, we introduce a discrete scheme and obtain optimal convergence results which are independent of the contrast of the coefficients. Numerical examples are presented to validate the theoretical results.

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Section: Introductionmentioning

confidence: 99%

A Nonconforming Virtual Element Method for the Elliptic Interface Problem

Wang,

Zheng,

null

et al. 2024

EAJAM

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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 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%

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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Virtual element methods for general linear elliptic interface problems on polygonal meshes with small edges

Cited by 15 publications

References 37 publications

A Nonconforming Virtual Element Method for the Elliptic Interface Problem

A Nonconforming Virtual Element Method for the Elliptic Interface Problem

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

A virtual element method for the elasticity problem allowing small edges

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