Volume 38, Issue 2
Error Analysis of a Stabilized Finite Element Method for the Generalized Stokes Problem

J. Comp. Math., 38 (2020), pp. 254-290.

Published online: 2020-02

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• Abstract

This paper is devoted to the establishment of sharper $a$ $priori$ stability and error estimates of a stabilized finite element method proposed by Barrenechea and Valentin for solving the generalized Stokes problem, which involves a viscosity $\nu$ and a reaction constant $\sigma$. With the establishment of sharper stability estimates and the help of $ad$ $hoc$ finite element projections, we can explicitly establish the dependence of error bounds of velocity and pressure on the viscosity $\nu$, the reaction constant $\sigma$, and the mesh size $h$. Our analysis reveals that the viscosity $\nu$ and the reaction constant $\sigma$ respectively act in the numerator position and the denominator position in the error estimates of velocity and pressure in standard norms without any weights. Consequently, the stabilization method is indeed suitable for the generalized Stokes problem with a small viscosity $\nu$ and a large reaction constant $\sigma$. The sharper error estimates agree very well with the numerical results.

• Keywords

Generalized Stokes equations, Stabilized finite element method, Error estimates.

65N12, 65N15, 65N30, 76M10

hyduan.math@whu.edu.cn (Huoyuan Duan)

scitance@nus.edu.sg (Roger C.E. Tan)

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@Article{JCM-38-254, author = {Duan , Huoyuan and Tan , Roger C.E. }, title = {Error Analysis of a Stabilized Finite Element Method for the Generalized Stokes Problem}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {2}, pages = {254--290}, abstract = {

This paper is devoted to the establishment of sharper $a$ $priori$ stability and error estimates of a stabilized finite element method proposed by Barrenechea and Valentin for solving the generalized Stokes problem, which involves a viscosity $\nu$ and a reaction constant $\sigma$. With the establishment of sharper stability estimates and the help of $ad$ $hoc$ finite element projections, we can explicitly establish the dependence of error bounds of velocity and pressure on the viscosity $\nu$, the reaction constant $\sigma$, and the mesh size $h$. Our analysis reveals that the viscosity $\nu$ and the reaction constant $\sigma$ respectively act in the numerator position and the denominator position in the error estimates of velocity and pressure in standard norms without any weights. Consequently, the stabilization method is indeed suitable for the generalized Stokes problem with a small viscosity $\nu$ and a large reaction constant $\sigma$. The sharper error estimates agree very well with the numerical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1805-m2017-0192}, url = {http://global-sci.org/intro/article_detail/jcm/14517.html} }
TY - JOUR T1 - Error Analysis of a Stabilized Finite Element Method for the Generalized Stokes Problem AU - Duan , Huoyuan AU - Tan , Roger C.E. JO - Journal of Computational Mathematics VL - 2 SP - 254 EP - 290 PY - 2020 DA - 2020/02 SN - 38 DO - http://doi.org/10.4208/jcm.1805-m2017-0192 UR - https://global-sci.org/intro/article_detail/jcm/14517.html KW - Generalized Stokes equations, Stabilized finite element method, Error estimates. AB -

This paper is devoted to the establishment of sharper $a$ $priori$ stability and error estimates of a stabilized finite element method proposed by Barrenechea and Valentin for solving the generalized Stokes problem, which involves a viscosity $\nu$ and a reaction constant $\sigma$. With the establishment of sharper stability estimates and the help of $ad$ $hoc$ finite element projections, we can explicitly establish the dependence of error bounds of velocity and pressure on the viscosity $\nu$, the reaction constant $\sigma$, and the mesh size $h$. Our analysis reveals that the viscosity $\nu$ and the reaction constant $\sigma$ respectively act in the numerator position and the denominator position in the error estimates of velocity and pressure in standard norms without any weights. Consequently, the stabilization method is indeed suitable for the generalized Stokes problem with a small viscosity $\nu$ and a large reaction constant $\sigma$. The sharper error estimates agree very well with the numerical results.

Huoyuan Duan & Roger C.E. Tan. (2020). Error Analysis of a Stabilized Finite Element Method for the Generalized Stokes Problem. Journal of Computational Mathematics. 38 (2). 254-290. doi:10.4208/jcm.1805-m2017-0192
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