Volume 40, Issue 5
Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations

J. Comp. Math., 40 (2022), pp. 732-759.

Published online: 2022-05

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• Abstract

Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper. These methods are based on the weakly continuous $P_1$ vector fields and the locally divergence-free (LDF) finite elements, which respectively penalize local divergence and are discontinuous across edges. These methods have no penalty factors and avoid solving the saddle-point problems. The existence and uniqueness of the velocity solution are proved, and the optimal error estimates of the energy norms and $L^2$-norms are obtained. Moreover, we propose unified pressure recovery algorithms and prove the optimal error estimates of $L^2$-norm for pressure. We design a unified iterative method for numerical experiments to verify the correctness of the theoretical analysis.

• Keywords

Stationary Navier-Stokes equations, Nonconforming nite elements, Penalty stabilization methods, DG methods, Locally divergence-free.

65N30, 76D05

982293064@qq.com (Linshuang He)

fmf@scu.edu.cn (Minfu Feng)

maqiang809@scu.edu.cn (Qiang Ma)

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@Article{JCM-40-732, author = {Linshuang and He and 982293064@qq.com and 23355 and School of Mathematics, Sichuan University, Chengdu 610064, China and Linshuang He and Minfu and Feng and fmf@scu.edu.cn and 6619 and School of Mathematics, Sichuan University, Chengdu 610064, China and Minfu Feng and Qiang and Ma and maqiang809@scu.edu.cn and 23356 and School of Mathematics, Sichuan University, Chengdu 610064, China and Qiang Ma}, title = {Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {5}, pages = {732--759}, abstract = {

Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper. These methods are based on the weakly continuous $P_1$ vector fields and the locally divergence-free (LDF) finite elements, which respectively penalize local divergence and are discontinuous across edges. These methods have no penalty factors and avoid solving the saddle-point problems. The existence and uniqueness of the velocity solution are proved, and the optimal error estimates of the energy norms and $L^2$-norms are obtained. Moreover, we propose unified pressure recovery algorithms and prove the optimal error estimates of $L^2$-norm for pressure. We design a unified iterative method for numerical experiments to verify the correctness of the theoretical analysis.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2101-m2020-0156}, url = {http://global-sci.org/intro/article_detail/jcm/20545.html} }
TY - JOUR T1 - Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations AU - He , Linshuang AU - Feng , Minfu AU - Ma , Qiang JO - Journal of Computational Mathematics VL - 5 SP - 732 EP - 759 PY - 2022 DA - 2022/05 SN - 40 DO - http://doi.org/10.4208/jcm.2101-m2020-0156 UR - https://global-sci.org/intro/article_detail/jcm/20545.html KW - Stationary Navier-Stokes equations, Nonconforming nite elements, Penalty stabilization methods, DG methods, Locally divergence-free. AB -

Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper. These methods are based on the weakly continuous $P_1$ vector fields and the locally divergence-free (LDF) finite elements, which respectively penalize local divergence and are discontinuous across edges. These methods have no penalty factors and avoid solving the saddle-point problems. The existence and uniqueness of the velocity solution are proved, and the optimal error estimates of the energy norms and $L^2$-norms are obtained. Moreover, we propose unified pressure recovery algorithms and prove the optimal error estimates of $L^2$-norm for pressure. We design a unified iterative method for numerical experiments to verify the correctness of the theoretical analysis.

Linshuang He, Minfu Feng & Qiang Ma. (2022). Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations. Journal of Computational Mathematics. 40 (5). 732-759. doi:10.4208/jcm.2101-m2020-0156
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