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Volume 23, Issue 3
A Weak Galerkin Finite Element Method for the Navier-Stokes Equations

Jiachuan Zhang, Kai Zhang, Jingzhi Li & Xiaoshen Wang

DOI: 10.4208/cicp.OA-2016-0267

Commun. Comput. Phys., 23 (2018), pp. 706-746.

Published online: 2018-03

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  • Abstract

In this paper, a weak Galerkin finite element method (WGFEM) is proposed for solving the Navier-Stokes equations (NSEs). The existence and uniqueness of the WGFEM solution of NSEs are established. The WGFEM provides very accurate numerical approximations for both the velocity field and pressure field, even with very high Reynolds numbers. The salient feature is that the flexibility of the WGFEM for the choice of the order of the velocity and pressure comparing to the standard finite element methods. Optimal order error estimates in both |·|$W_h$ and $L^2$ norms are proved for the semi-discretized scheme. Numerical simulations are presented to show the efficiency of the WGFEM.

  • Keywords

Weak Galerkin, finite element method, the Navier-Stokes equations.

  • AMS Subject Headings

65N15, 65N30, 65M12, 76D07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-23-706, author = {Jiachuan Zhang, Kai Zhang, Jingzhi Li and Xiaoshen Wang}, title = {A Weak Galerkin Finite Element Method for the Navier-Stokes Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {23}, number = {3}, pages = {706--746}, abstract = {

In this paper, a weak Galerkin finite element method (WGFEM) is proposed for solving the Navier-Stokes equations (NSEs). The existence and uniqueness of the WGFEM solution of NSEs are established. The WGFEM provides very accurate numerical approximations for both the velocity field and pressure field, even with very high Reynolds numbers. The salient feature is that the flexibility of the WGFEM for the choice of the order of the velocity and pressure comparing to the standard finite element methods. Optimal order error estimates in both |·|$W_h$ and $L^2$ norms are proved for the semi-discretized scheme. Numerical simulations are presented to show the efficiency of the WGFEM.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0267}, url = {http://global-sci.org/intro/article_detail/cicp/10545.html} }
TY - JOUR T1 - A Weak Galerkin Finite Element Method for the Navier-Stokes Equations AU - Jiachuan Zhang, Kai Zhang, Jingzhi Li & Xiaoshen Wang JO - Communications in Computational Physics VL - 3 SP - 706 EP - 746 PY - 2018 DA - 2018/03 SN - 23 DO - http://doi.org/10.4208/cicp.OA-2016-0267 UR - https://global-sci.org/intro/article_detail/cicp/10545.html KW - Weak Galerkin, finite element method, the Navier-Stokes equations. AB -

In this paper, a weak Galerkin finite element method (WGFEM) is proposed for solving the Navier-Stokes equations (NSEs). The existence and uniqueness of the WGFEM solution of NSEs are established. The WGFEM provides very accurate numerical approximations for both the velocity field and pressure field, even with very high Reynolds numbers. The salient feature is that the flexibility of the WGFEM for the choice of the order of the velocity and pressure comparing to the standard finite element methods. Optimal order error estimates in both |·|$W_h$ and $L^2$ norms are proved for the semi-discretized scheme. Numerical simulations are presented to show the efficiency of the WGFEM.

Jiachuan Zhang, Kai Zhang, Jingzhi Li and Xiaoshen Wang. (2018). A Weak Galerkin Finite Element Method for the Navier-Stokes Equations. Communications in Computational Physics. 23 (3). 706-746. doi:10.4208/cicp.OA-2016-0267
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