Symmetric and Nonsymmetric Discontinuous Galerkin Methods for a Pseudostress Formulation of the Stokes Spectral Problem

Lepe, Felipe; Mora, David

doi:10.1137/19m1259535

Cited by 22 publications

(13 citation statements)

References 22 publications

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“…Nevertheless, the main disadvantage of this method lies in the correct choice of the stabilization parameter, since, depending on the configuration of the problem, namely the geometry, boundary conditions or physical quantities, it can generate spurious eigenvalues in the computed spectrum. This has been also observed in other problems where the methods need to be stabilized for some parameter, as it occurs in [2,19,20,24], just for mention some recent papers that deal with this subject.…”

Section: Introductionmentioning

confidence: 58%

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Displacement-pseudostress formulation for the linear elasticity spectral problem

Inzunza¹,

Lepe²,

Rivera³

2021

Preprint

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In this paper we analyze a mixed displacement-pseudostress formulation for the elasticity eigenvalue problem. We propose a finite element method to approximate the pseudostress tensor with Raviart-Thomas elements and the displacement with piecewise polynomials. With the aid of the classic theory for compact operators, we prove that our method is convergent and does not introduce spurious modes. Also, we obtain error estimates for the proposed method. Finally, we report some numerical tests supporting the theoretical results.

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Section: Introductionmentioning

confidence: 58%

“…Clearly, with this test we are considering a particular case where we will approximate a curved domain with triangles. This geometrical features will be reflected in the order of convergence, as it happens, for instance, in [19] for the DG method.…”

Section: Unitary Cubementioning

confidence: 98%

Displacement-pseudostress formulation for the linear elasticity spectral problem

Inzunza¹,

Lepe²,

Rivera³

2021

Preprint

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“…In fact, we notice that for k = 0, the order of approximation is clearly h 2 , meanwhile for k > 0 the observed order is close to h 2(k+1) . This increased order is expectable for high order methods, as it happen, for example, in DG methods (see for instance [8,18]). In Table 2, the results from using the [P k ] n -RT k scheme are provided.…”

Section: Numerical Experimentsmentioning

confidence: 86%

Mixed methods for the velocity-pressure-pseudostress formulation of the Stokes eigenvalue problem

Lepe¹,

Rivera²,

Vellojin³

2021

Preprint

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In two and three dimensional domains, we analyze mixed finite element methods for a velocity-pressure-pseudostress formulation of the Stokes eigenvalue problem. The methods consist in two schemes: the velocity and pressure are approximated with piecewise polynomial and 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. In order to assess the performance of the schemes, we report numerical results to compare the accuracy and robustness between both numerical schemes.

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“…The pseudostress formulation has been recently analyzed in [25], with different DG methods based in interior penalization. In this methods, the stabilization parameter affects strongly the behavior of spurious eigenvalues and the choice of such parameter, in order to avoid the spurious eigenvalues, depends on the configuration of the problem, namely the geometry and boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

A virtual element approximation for the pseudostress formulation of the Stokes eigenvalue problem

Lepe,

Rivera

2020

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In this paper we analyze a virtual element method (VEM) for a pseudostress formulation of the Stokes eigenvalue problem. This formulation allows to eliminate the velocity and the pressure, leading to an elliptic formulation where the only unknown is the pseudostress tensor. The velocity and pressure can be recovered by a post-process. Adapting the non-compact operator theory, we prove that our method provides a correct approximation of the spectrum and is spurious free. We prove a priori error estimates, with optimal order, which we confirm with some numerical tests.

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Symmetric and Nonsymmetric Discontinuous Galerkin Methods for a Pseudostress Formulation of the Stokes Spectral Problem

Cited by 22 publications

References 22 publications

Displacement-pseudostress formulation for the linear elasticity spectral problem

Displacement-pseudostress formulation for the linear elasticity spectral problem

Mixed methods for the velocity-pressure-pseudostress formulation of the Stokes eigenvalue problem

A virtual element approximation for the pseudostress formulation of the Stokes eigenvalue problem

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