Analysis of the small viscosity and large reaction coefficient in the computation of the generalized Stokes problem by a novel stabilized finite element method

Huo-Yuan Duan Wuhan University Po-Wen Hsieh Chung Yuan Christian University Roger Choon Ee Tan National University of Singapore Suh-Yuh Yang National Central University

Numerical Analysis and Scientific Computing mathscidoc:1703.25034

Computer Methods in Applied Mechanics and Engineering, 271, 2014.4
In this paper, we propose and analyze a novel stabilized finite element method (FEM) for the system of generalized Stokes equations arising from the time-discretization of transient Stokes problem. The system involves a small viscosity, which is proportional to the inverse of large Reynolds number, and a large reaction coefficient, which is the inverse of small time step. The proposed stabilized FEM employs the $C^0$ piecewise linear elements for both velocity field and pressure on the same mesh and uses the residuals of the momentum equation and the divergence-free equation to define the stabilization terms. The stabilization parameters are fixed and element-independent, without a comparison of the viscosity, the reaction coefficient and the mesh size. Using the finite element solution of an auxiliary boundary value problem as the interpolating function for velocity and the $H^1$-seminorm projection for pressure, instead of the usual nodal interpolants, we derive error estimates for the stabilized finite element approximations to velocity and pressure in the $L^2$ and $H^1$ norms and most importantly, we explicitly establish the dependence of error bounds on the viscosity, the reaction coefficient and the mesh size. Our analysis reveals that this stabilized FEM is particularly suitable for the generalized Stokes system with a small viscosity and a large reaction coefficient, which has never been achieved before in the error analysis of other stabilization methods in the literature. We then numerically confirm the effectiveness of the proposed stabilized FEM. Comparisons made with other existing stabilization methods show that the newly proposed method can attain better accuracy and stability.
generalized Stokes problem, small viscosity, large reaction coefficient, stabilized finite element method, stabilization parameter
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@inproceedings{huo-yuan2014analysis,
  title={Analysis of the small viscosity and large reaction coefficient in the computation of the generalized Stokes problem by a novel stabilized finite element method},
  author={Huo-Yuan Duan, Po-Wen Hsieh, Roger Choon Ee Tan, and Suh-Yuh Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170314205509373936707},
  booktitle={Computer Methods in Applied Mechanics and Engineering},
  volume={271},
  year={2014},
}
Huo-Yuan Duan, Po-Wen Hsieh, Roger Choon Ee Tan, and Suh-Yuh Yang. Analysis of the small viscosity and large reaction coefficient in the computation of the generalized Stokes problem by a novel stabilized finite element method. 2014. Vol. 271. In Computer Methods in Applied Mechanics and Engineering. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170314205509373936707.
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