Volume 5, Issue 1
Two-Level Newton Iteration Methods for Navier-Stokes Type Variational Inequality Problem

Rong An & Hailong Qiu

Adv. Appl. Math. Mech., 5 (2013), pp. 36-54.

Published online: 2013-05

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  • Abstract

This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem. We solve a small Navier-Stokes problem on the coarse mesh with mesh size $H$ and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size $h$. The error estimates derived show that if we choose $h=\mathcal{O}(|\log h|^{1/2}H^3)$, then the two-level method we provide has the same $H^1$ and $L^2$ convergence orders of the velocity and the pressure as the one-level stabilized method. However, the $L^2$ convergence order of the velocity is not consistent with that of one-level stabilized method. Finally, we give the numerical results to support the theoretical analysis.

  • Keywords

Navier-Stokes equations, nonlinear slip boundary conditions, variational inequality problem, stabilized finite element, two-level methods.

  • AMS Subject Headings

65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-5-36, author = {Rong An , and Qiu , Hailong}, title = {Two-Level Newton Iteration Methods for Navier-Stokes Type Variational Inequality Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2013}, volume = {5}, number = {1}, pages = {36--54}, abstract = {

This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem. We solve a small Navier-Stokes problem on the coarse mesh with mesh size $H$ and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size $h$. The error estimates derived show that if we choose $h=\mathcal{O}(|\log h|^{1/2}H^3)$, then the two-level method we provide has the same $H^1$ and $L^2$ convergence orders of the velocity and the pressure as the one-level stabilized method. However, the $L^2$ convergence order of the velocity is not consistent with that of one-level stabilized method. Finally, we give the numerical results to support the theoretical analysis.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.11-m11188}, url = {http://global-sci.org/intro/article_detail/aamm/56.html} }
TY - JOUR T1 - Two-Level Newton Iteration Methods for Navier-Stokes Type Variational Inequality Problem AU - Rong An , AU - Qiu , Hailong JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 36 EP - 54 PY - 2013 DA - 2013/05 SN - 5 DO - http://doi.org/10.4208/aamm.11-m11188 UR - https://global-sci.org/intro/article_detail/aamm/56.html KW - Navier-Stokes equations, nonlinear slip boundary conditions, variational inequality problem, stabilized finite element, two-level methods. AB -

This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem. We solve a small Navier-Stokes problem on the coarse mesh with mesh size $H$ and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size $h$. The error estimates derived show that if we choose $h=\mathcal{O}(|\log h|^{1/2}H^3)$, then the two-level method we provide has the same $H^1$ and $L^2$ convergence orders of the velocity and the pressure as the one-level stabilized method. However, the $L^2$ convergence order of the velocity is not consistent with that of one-level stabilized method. Finally, we give the numerical results to support the theoretical analysis.

Rong An & Hailong Qiu. (1970). Two-Level Newton Iteration Methods for Navier-Stokes Type Variational Inequality Problem. Advances in Applied Mathematics and Mechanics. 5 (1). 36-54. doi:10.4208/aamm.11-m11188
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