Volume 6, Issue 5
Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow

Adv. Appl. Math. Mech., 6 (2014), pp. 663-679.

Published online: 2014-06

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• Abstract

In this paper, we consider a two-scale stabilized finite volume method for the two-dimensional stationary incompressible flow approximated by the lowest equal-order element pair $P_1-P_1$ which do not satisfy the inf-sup condition. The two-scale method consist of solving a small non-linear system on the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimal order in the $H^1$-norm for velocity and the $L^2$-norm for pressure are obtained. The error analysis shows there is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relation  $h =\mathcal{O}(H^2)$.  Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.

• Keywords

Incompressible flow stabilized finite volume method {inf-sup} condition local Gauss integral two-scale method

35Q10 65N30 76D05

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@Article{AAMM-6-663, author = {Jianhong Yang, Gang Lei and Jianwei Yang}, title = {Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2014}, volume = {6}, number = {5}, pages = {663--679}, abstract = {

In this paper, we consider a two-scale stabilized finite volume method for the two-dimensional stationary incompressible flow approximated by the lowest equal-order element pair $P_1-P_1$ which do not satisfy the inf-sup condition. The two-scale method consist of solving a small non-linear system on the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimal order in the $H^1$-norm for velocity and the $L^2$-norm for pressure are obtained. The error analysis shows there is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relation  $h =\mathcal{O}(H^2)$.  Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2013.m153}, url = {http://global-sci.org/intro/article_detail/aamm/41.html} }
TY - JOUR T1 - Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow AU - Jianhong Yang, Gang Lei & Jianwei Yang JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 663 EP - 679 PY - 2014 DA - 2014/06 SN - 6 DO - http://doi.org/10.4208/aamm.2013.m153 UR - https://global-sci.org/intro/article_detail/aamm/41.html KW - Incompressible flow KW - stabilized finite volume method KW - {inf-sup} condition KW - local Gauss integral KW - two-scale method AB -

In this paper, we consider a two-scale stabilized finite volume method for the two-dimensional stationary incompressible flow approximated by the lowest equal-order element pair $P_1-P_1$ which do not satisfy the inf-sup condition. The two-scale method consist of solving a small non-linear system on the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimal order in the $H^1$-norm for velocity and the $L^2$-norm for pressure are obtained. The error analysis shows there is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relation  $h =\mathcal{O}(H^2)$.  Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.

Jianhong Yang, Gang Lei & Jianwei Yang. (1970). Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow. Advances in Applied Mathematics and Mechanics. 6 (5). 663-679. doi:10.4208/aamm.2013.m153
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