Volume 5, Issue 1
Two-Level Stabilized Finite Volume Methods for the Stationary Navier-Stokes Equations

Adv. Appl. Math. Mech., 5 (2013), pp. 19-35.

Published online: 2013-05

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• Abstract

In this work, two-level stabilized finite volume formulations for the 2D steady Navier-Stokes equations are considered. These methods are based on the local Gauss integration technique and the lowest equal-order finite element pair. Moreover, the two-level stabilized finite volume methods involve solving one small Navier-Stokes problem on a coarse mesh with mesh size $H$, a large general Stokes problem for the Simple and Oseen two-level stabilized finite volume methods on the fine mesh with mesh size $h$=$\mathcal{O}(H^2)$ or a large general Stokes equations for the Newton two-level stabilized finite volume method on a fine mesh with mesh size $h$=$\mathcal{O}(|\log h|^{1/2}H^3)$. These methods we studied provide an approximate solution $(\widetilde{u}_h^v,\widetilde{p}_h^v)$ with the convergence rate of same order as the standard stabilized finite volume method, which involve solving one large nonlinear problem on a fine mesh with mesh size $h$. Hence, our methods can save a large amount of computational time.

• Keywords

Stationary Navier-Stokes equations, finite volume method, two-level method, error estimate.

65N30, 65N08, 76D05

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@Article{AAMM-5-19, author = {Tong Zhang , and Xu , Shunwei}, title = {Two-Level Stabilized Finite Volume Methods for the Stationary Navier-Stokes Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2013}, volume = {5}, number = {1}, pages = {19--35}, abstract = {

In this work, two-level stabilized finite volume formulations for the 2D steady Navier-Stokes equations are considered. These methods are based on the local Gauss integration technique and the lowest equal-order finite element pair. Moreover, the two-level stabilized finite volume methods involve solving one small Navier-Stokes problem on a coarse mesh with mesh size $H$, a large general Stokes problem for the Simple and Oseen two-level stabilized finite volume methods on the fine mesh with mesh size $h$=$\mathcal{O}(H^2)$ or a large general Stokes equations for the Newton two-level stabilized finite volume method on a fine mesh with mesh size $h$=$\mathcal{O}(|\log h|^{1/2}H^3)$. These methods we studied provide an approximate solution $(\widetilde{u}_h^v,\widetilde{p}_h^v)$ with the convergence rate of same order as the standard stabilized finite volume method, which involve solving one large nonlinear problem on a fine mesh with mesh size $h$. Hence, our methods can save a large amount of computational time.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.11-m11178}, url = {http://global-sci.org/intro/article_detail/aamm/55.html} }
TY - JOUR T1 - Two-Level Stabilized Finite Volume Methods for the Stationary Navier-Stokes Equations AU - Tong Zhang , AU - Xu , Shunwei JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 19 EP - 35 PY - 2013 DA - 2013/05 SN - 5 DO - http://doi.org/10.4208/aamm.11-m11178 UR - https://global-sci.org/intro/article_detail/aamm/55.html KW - Stationary Navier-Stokes equations, finite volume method, two-level method, error estimate. AB -

In this work, two-level stabilized finite volume formulations for the 2D steady Navier-Stokes equations are considered. These methods are based on the local Gauss integration technique and the lowest equal-order finite element pair. Moreover, the two-level stabilized finite volume methods involve solving one small Navier-Stokes problem on a coarse mesh with mesh size $H$, a large general Stokes problem for the Simple and Oseen two-level stabilized finite volume methods on the fine mesh with mesh size $h$=$\mathcal{O}(H^2)$ or a large general Stokes equations for the Newton two-level stabilized finite volume method on a fine mesh with mesh size $h$=$\mathcal{O}(|\log h|^{1/2}H^3)$. These methods we studied provide an approximate solution $(\widetilde{u}_h^v,\widetilde{p}_h^v)$ with the convergence rate of same order as the standard stabilized finite volume method, which involve solving one large nonlinear problem on a fine mesh with mesh size $h$. Hence, our methods can save a large amount of computational time.

Tong Zhang & Shunwei Xu. (1970). Two-Level Stabilized Finite Volume Methods for the Stationary Navier-Stokes Equations. Advances in Applied Mathematics and Mechanics. 5 (1). 19-35. doi:10.4208/aamm.11-m11178
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