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Volume 26, Issue 3
Finite Element Methods for the Navier-Stokes Equations by $H(div)$ Elements

Junping Wang, Xiaoshen Wang & Xiu Ye

J. Comp. Math., 26 (2008), pp. 410-436.

Published online: 2008-06

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  • Abstract

We derived and analyzed a new numerical scheme for the Navier-Stokes equations by using $H(div)$ conforming finite elements. A great deal of effort was given to an establishment of some Sobolev-type inequalities for piecewise smooth functions. In particular, the newly derived Sobolev inequalities were employed to provide a mathematical theory for the $H(div)$ finite element scheme. For example, it was proved that the new finite element scheme has solutions which admit a certain boundedness in terms of the input data. A solution uniqueness was also possible when the input data satisfies a certain smallness condition. Optimal-order error estimates for the corresponding finite element solutions were established in various Sobolev norms. The finite element solutions from the new scheme feature a full satisfaction of the continuity equation which is highly demanded in scientific computing.

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@Article{JCM-26-410, author = {}, title = {Finite Element Methods for the Navier-Stokes Equations by $H(div)$ Elements}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {3}, pages = {410--436}, abstract = {

We derived and analyzed a new numerical scheme for the Navier-Stokes equations by using $H(div)$ conforming finite elements. A great deal of effort was given to an establishment of some Sobolev-type inequalities for piecewise smooth functions. In particular, the newly derived Sobolev inequalities were employed to provide a mathematical theory for the $H(div)$ finite element scheme. For example, it was proved that the new finite element scheme has solutions which admit a certain boundedness in terms of the input data. A solution uniqueness was also possible when the input data satisfies a certain smallness condition. Optimal-order error estimates for the corresponding finite element solutions were established in various Sobolev norms. The finite element solutions from the new scheme feature a full satisfaction of the continuity equation which is highly demanded in scientific computing.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10360.html} }
TY - JOUR T1 - Finite Element Methods for the Navier-Stokes Equations by $H(div)$ Elements JO - Journal of Computational Mathematics VL - 3 SP - 410 EP - 436 PY - 2008 DA - 2008/06 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10360.html KW - Finite element methods, Navier-Stokes equations, CFD. AB -

We derived and analyzed a new numerical scheme for the Navier-Stokes equations by using $H(div)$ conforming finite elements. A great deal of effort was given to an establishment of some Sobolev-type inequalities for piecewise smooth functions. In particular, the newly derived Sobolev inequalities were employed to provide a mathematical theory for the $H(div)$ finite element scheme. For example, it was proved that the new finite element scheme has solutions which admit a certain boundedness in terms of the input data. A solution uniqueness was also possible when the input data satisfies a certain smallness condition. Optimal-order error estimates for the corresponding finite element solutions were established in various Sobolev norms. The finite element solutions from the new scheme feature a full satisfaction of the continuity equation which is highly demanded in scientific computing.

Junping Wang, Xiaoshen Wang & Xiu Ye. (1970). Finite Element Methods for the Navier-Stokes Equations by $H(div)$ Elements. Journal of Computational Mathematics. 26 (3). 410-436. doi:
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