TY - JOUR

T1 - Analysis of the small viscosity and large reaction coefficient in the computation of the generalized Stokes problem by a novel stabilized finite element method

AU - Duan, Huo Yuan

AU - Hsieh, Po Wen

AU - Tan, Roger C.E.

AU - Yang, Suh Yuh

N1 - Funding Information:
The work of H.-Y. Duan was partially supported by the National Natural Science Foundation of China under the Grants 11071132 and 11171168 and the Research Fund for the Doctoral Program of Higher Education of China under Grants 20100031110002 and 20120031110026 . The work of P.-W. Hsieh and S.-Y. Yang was partially supported by the National Science Council of Taiwan under the Grants NSC 102-2115-M-033-007-MY2 and NSC 101-2115-M-008-008-MY2 .

PY - 2014/4/1

Y1 - 2014/4/1

N2 - 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 C0 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 H1-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 L2 and H1 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.

AB - 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 C0 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 H1-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 L2 and H1 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.

KW - Generalized Stokes problem

KW - Large reaction coefficient

KW - Small viscosity

KW - Stabilization parameter

KW - Stabilized finite element method

UR - http://www.scopus.com/inward/record.url?scp=84891638322&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2013.11.024

DO - 10.1016/j.cma.2013.11.024

M3 - 期刊論文

AN - SCOPUS:84891638322

SN - 0045-7825

VL - 271

SP - 23

EP - 47

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

ER -