Year: 2020
Author: Huoyuan Duan, Roger C.E. Tan
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 2 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1805-m2017-0192
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 2 : pp. 254–290
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 37
Keywords: Generalized Stokes equations Stabilized finite element method Error estimates.