A posteriori error estimates for a Virtual Element Method for the Steklov eigenvalue problem

Mora, David; Rivera, Gonzalo; Rodrı́guez, Rodolfo

doi:10.1016/j.camwa.2017.05.016

Cited by 54 publications

(30 citation statements)

References 42 publications

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“…They are also compact since their ranges are finite dimensional. , u) is an eigenpair for the operator T , and analogously for problems (41) and (42) and operators T h and T h . By virtue of this correspondence, the convergence analysis can be derived from the spectral approximation theory for compact operators.…”

Section: Spectral Approximation For Compact Operatorsmentioning

confidence: 99%

“…We remark that the conforming VEM formulation has been proposed for the approximation of the Steklov eigenvalue problem [40,41], the Laplace eigenvalue problem [34], the acoustic vibration problem [13], and the vibration problem of Kirchhoff plates [42], whereas [24] deals with the Mimetic Finite Difference approximation of the eigenvalue problem in mixed form.…”

Section: Introductionmentioning

confidence: 99%

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The nonconforming Virtual Element Method for eigenvalue problems

2019

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We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two-and three-dimensional case. We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L 2 -inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

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Section: Spectral Approximation For Compact Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The nonconforming Virtual Element Method for eigenvalue problems

2019

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“…The a posteriori version of VEM was investigated in [17,20,26,27,46] but the hp adaptivity has never been targeted before and is the topic of the present work. This paper falls in a series of articles covering different aspects of the p and hp version of VEM, namely a priori error analysis on quasi-uniform meshes [11], approximation of corner singularities [12], multigrid algorithms [5], ill-conditioning in two [39] and three dimensional problems [31], Trefftz and non-conforming approaches [29,40], theoretical and numerical analysis of the stabilization typical of VEM [38].…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimation and adaptivity in hp virtual elements

2019

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An explicit and computable error estimator for the hp version of the virtual element method (VEM), together with lower and upper bounds with respect to the exact energy error, is presented. Such error estimator is employed to provide, following the approach of [42], hp adaptive mesh refinements for very general polygonal meshes. In addition, a novel VEM hp Clément quasi-interpolant, instrumental for the a posteriori error analysis, is introduced. The performances of the adaptive method are validated by a number of numerical experiments.

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“…Although the VEM is very recent, it has been applied to a large number of problems; for instance, VEM for Stokes, Brinkman, Cahn-Hilliard, plates bending, advection-diffusion, Helmholtz, parabolic, and hyperbolic problems have been introduced in [4,5,15,17,24,19,21,26,27,30,51,54,55,56], VEM for spectral problems in [18,37,42,44], VEM for linear and non-linear elasticity in [6,9,13,36,57], whereas a posteriori error analysis have been developed in [16,20,28,43].…”

Section: Introductionmentioning

confidence: 99%

A virtual element method for a nonlocal FitzHugh–Nagumo model of cardiac electrophysiology

Anaya

Bendahmane

Mora

et al. 2019

IMA Journal of Numerical Analysis

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We present a Virtual Element Method (VEM) for a nonlocal reaction-diffusion system of the cardiac electric field. To this system, we analyze an H 1 (Ω)-conforming discretization by means of VEM which can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general L p compactness criterion. Moreover, we obtain optimal order space-time error estimates in the L 2 norm. Finally, we report some numerical tests supporting the theoretical results.

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A posteriori error estimates for a Virtual Element Method for the Steklov eigenvalue problem

Cited by 54 publications

References 42 publications

The nonconforming Virtual Element Method for eigenvalue problems

The nonconforming Virtual Element Method for eigenvalue problems

A posteriori error estimation and adaptivity in hp virtual elements

A virtual element method for a nonlocal FitzHugh–Nagumo model of cardiac electrophysiology

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