An analysis of the Rayleigh–Stokes problem for a generalized second-grade fluid

Emilia Bazhlekova; Bangti Jin; Raytcho Lazarov; Zhi Zhou

doi:10.1007/s00211-014-0685-2

An analysis of the Rayleigh–Stokes problem for a generalized second-grade fluid

Emilia Bazhlekova, Bangti Jin, Raytcho Lazarov, Zhi Zhou

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Abstract

© 2014, The Author(s). We study the Rayleigh–Stokes problem for a generalized second-grade fluid which involves a Riemann–Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete formulations. We establish the Sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v, including v∈$^{L2}$(Ω). A space semidiscrete Galerkin scheme using continuous piecewise linear finite elements is developed, and optimal with respect to initial data regularity error estimates for the finite element approximations are derived. Further, two fully discrete schemes based on the backward Euler method and second-order backward difference method and the related convolution quadrature are developed, and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data. Numerical results for one- and two-dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method, and to verify the convergence theory.

Original language	English (US)
Pages (from-to)	1-31
Number of pages	31
Journal	Numerische Mathematik
Volume	131
Issue number	1
DOIs	https://doi.org/10.1007/s00211-014-0685-2
State	Published - Nov 26 2014
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: The authors are grateful to Prof. Christian Lubich for his helpful comments on an earlier version of the paper, which led to a significant improvement of the presentation in Sect. 4, and an anonymous referee for many constructive comments. The research of B. Jin has been supported by NSF Grant DMS-1319052, and R. Lazarov was supported in parts by NSF Grant DMS-1016525 and also by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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abstract = "{\textcopyright} 2014, The Author(s). We study the Rayleigh–Stokes problem for a generalized second-grade fluid which involves a Riemann–Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete formulations. We establish the Sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v, including v∈$^{L2}$(Ω). A space semidiscrete Galerkin scheme using continuous piecewise linear finite elements is developed, and optimal with respect to initial data regularity error estimates for the finite element approximations are derived. Further, two fully discrete schemes based on the backward Euler method and second-order backward difference method and the related convolution quadrature are developed, and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data. Numerical results for one- and two-dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method, and to verify the convergence theory.",

author = "Emilia Bazhlekova and Bangti Jin and Raytcho Lazarov and Zhi Zhou",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUS-C1-016-04 Acknowledgements: The authors are grateful to Prof. Christian Lubich for his helpful comments on an earlier version of the paper, which led to a significant improvement of the presentation in Sect. 4, and an anonymous referee for many constructive comments. The research of B. Jin has been supported by NSF Grant DMS-1319052, and R. Lazarov was supported in parts by NSF Grant DMS-1016525 and also by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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N2 - © 2014, The Author(s). We study the Rayleigh–Stokes problem for a generalized second-grade fluid which involves a Riemann–Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete formulations. We establish the Sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v, including v∈$^{L2}$(Ω). A space semidiscrete Galerkin scheme using continuous piecewise linear finite elements is developed, and optimal with respect to initial data regularity error estimates for the finite element approximations are derived. Further, two fully discrete schemes based on the backward Euler method and second-order backward difference method and the related convolution quadrature are developed, and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data. Numerical results for one- and two-dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method, and to verify the convergence theory.

AB - © 2014, The Author(s). We study the Rayleigh–Stokes problem for a generalized second-grade fluid which involves a Riemann–Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete formulations. We establish the Sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v, including v∈$^{L2}$(Ω). A space semidiscrete Galerkin scheme using continuous piecewise linear finite elements is developed, and optimal with respect to initial data regularity error estimates for the finite element approximations are derived. Further, two fully discrete schemes based on the backward Euler method and second-order backward difference method and the related convolution quadrature are developed, and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data. Numerical results for one- and two-dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method, and to verify the convergence theory.

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An analysis of the Rayleigh–Stokes problem for a generalized second-grade fluid

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