Chebyshev spectral collocation method approximations of the Stokes eigenvalue problem based on penalty techniques

Türk, Önder; Codina, Ramon

doi:10.1016/j.apnum.2019.06.005

Cited by 11 publications

(3 citation statements)

References 28 publications

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“…For the case of

Ha&#x0003D;0

, the smallest eigenvalue

{\lambda}_{1,h}&#x0003D;52.344691168

is presented in Gedicke and Khan, 33 and

{\lambda}_{1,h}&#x0003D;52.34469138411319

is provided in Türk and Codina 34 while the approximation we calculated by using

{\mathbb{P}}_3-{\mathbb{P}}_2

element with

dof&#x0003D;851968

{\lambda}_{1,h}&#x0003D;52.3446911687887

. These three values have the same six decimal places.…”

Section: Numerical Experimentsmentioning

confidence: 99%

The discontinuous Galerkin and the nonconforming ECR element approximations for an MHD Stokes eigenvalue problem

Sun

Yang

2022

Math Methods in App Sciences

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In this paper, we adopt the discontinuous Galerkin finite element method and the enriched Crouzeix‐Raviart finite element method to study the magnetohydrodynamic (MHD) Stokes eigenvalue problem describing the flow of a viscous and electrically conducting fluid in a duct under the influence of a uniform magnetic field. We give the convergence and error analysis for the approximations, and the theoretical analysis and numerical experiments show that the methods are effective and can be applied to general domains. We also explore the influence of the Hartmann number on the eigenpairs and the consequential variation of the eigenstructure with the magnetic field by numerical experiments.

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“…For the case of

Ha&#x0003D;0

, the smallest eigenvalue

{\lambda}_{1,h}&#x0003D;52.344691168

is presented in Gedicke and Khan, 33 and

{\lambda}_{1,h}&#x0003D;52.34469138411319

is provided in Türk and Codina 34 while the approximation we calculated by using

{\mathbb{P}}_3-{\mathbb{P}}_2

element with

dof&#x0003D;851968

{\lambda}_{1,h}&#x0003D;52.3446911687887

. These three values have the same six decimal places.…”

Section: Numerical Experimentsmentioning

confidence: 99%

The discontinuous Galerkin and the nonconforming ECR element approximations for an MHD Stokes eigenvalue problem

Sun

Yang

2022

Math Methods in App Sciences

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“…There are six classes of Chebyshev polynomials: first, second, third, fourth, fifth, and sixth kinds (Masjed-Jamei 2006). There are many old and recent studies interested in these polynomials (Abd-Elhameed et al 2016;Türk and Codina 2019;Abd-Elhameed andYoussri 2018, 2019;Abd-Elhameed 2021;Abd-Elhameed and Alkhamisi 2021;Atta et al 2022a). In this paper, our main focus is on the first type of Chebyshev polynomials and their shifted ones.…”

Section: Introductionmentioning

confidence: 99%

Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel

Atta

Youssri

2022

Comp. Appl. Math.

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This research apparatuses an approximate spectral method for the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel (TFPIDE). The main idea of this approach is to set up a new Hilbert space that satisfies the initial and boundary conditions. The new spectral collocation approach is applied to obtain precise numerical approximation using new basis functions based on shifted first-kind Chebyshev polynomials (SCP1K). Furthermore, we support our study by a careful error analysis of the suggested shifted first-kind Chebyshev expansion. The results show that the new approach is very accurate and effective.

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“…Tau method is more flexible than the Galerkin method due to the freedom of choosing the used basis functions, see for example [5,9,17,19]. The collocation method is common in the application as it can be applied to all kinds of differential equations, see for example [1,12,13,18,23,27,38].…”

Section: Introductionmentioning

confidence: 99%

New formulas of thehigh‐orderderivatives offifth‐kindChebyshev polynomials: Spectral solution of theconvection–diffusionequation

Abd-Elhameed

Youssri

2021

Numerical Methods Partial

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This paper is dedicated to deriving novel formulae for the high‐order derivatives of Chebyshev polynomials of the fifth‐kind. The high‐order derivatives of these polynomials are expressed in terms of their original polynomials. The derived formulae contain certain terminating 4F3(1) hypergeometric functions. We show that the resulting hypergeometric functions can be reduced in the case of the first derivative. As an important application—and based on the derived formulas—a spectral tau algorithm is implemented and analyzed for numerically solving the convection–diffusion equation. The convergence and error analysis of the suggested double expansion is investigated assuming that the solution of the problem is separable. Some illustrative examples are presented to check the applicability and accuracy of our proposed algorithm.

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Chebyshev spectral collocation method approximations of the Stokes eigenvalue problem based on penalty techniques

Cited by 11 publications

References 28 publications

The discontinuous Galerkin and the nonconforming ECR element approximations for an MHD Stokes eigenvalue problem

The discontinuous Galerkin and the nonconforming ECR element approximations for an MHD Stokes eigenvalue problem

Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel

New formulas of thehigh‐orderderivatives offifth‐kindChebyshev polynomials: Spectral solution of theconvection–diffusionequation

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