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Volume 12, Issue 2
Two-Grid Stabilized FEMs Based on Newton Type Linearization for the Steady-State Natural Convection Problem

Yunbo Yang, Yaolin Jiang & Qiongxiang Kong

Adv. Appl. Math. Mech., 12 (2020), pp. 407-435.

Published online: 2020-01

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  • Abstract

This paper is concerned with two types of two-grid stabilized finite element methods (FEMs) based on Newton iteration for the steady-state nature convection problem. The first method needs to solve one small nonlinear natural convection system on the coarse mesh with mesh width $H$, and then to solve one large linearized natural convection system on the fine mesh with mesh width $h=\mathcal {O}(H^2)$ based on Newton iteration. The other method needs to solve one small nonlinear natural convection system on the same coarse mesh, and then to solve two large linearized systems on the fine mesh with mesh width $h=\mathcal {O}(H^{\frac{7-\varepsilon}{2}})$ based on Newton iteration which have the same stiffness matrix with only different right-hand side. In both methods, the stabilization terms are defined via two local Gauss integrations at element level which has no need to introduce additional variables comparing with the standard variational multiscale stabilized FEMs. The stability estimates and the convergence analysis for both methods are derived strictly. Ample numerical results are presented to confirm the theoretical predictions and demonstrate the efficiency of the new methods.

  • Keywords

Natural convection problem, two-grid method, finite element methods, Newton iteration, variational multiscale stabilization, error estimates.

  • AMS Subject Headings

65N15, 65N30, 65N12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

ybyang13@126.com (Yunbo Yang)

  • BibTex
  • RIS
  • TXT
@Article{AAMM-12-407, author = {Yunbo and Yang and ybyang13@126.com and 6481 and Department of Mathematics, Yunnan Normal University, Kunming 650500, Yunnan, China and Yunbo Yang and Yaolin and Jiang and and 6482 and School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China and Yaolin Jiang and Qiongxiang and Kong and and 6483 and School of Human Settlements and Civil Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China and Qiongxiang Kong}, title = {Two-Grid Stabilized FEMs Based on Newton Type Linearization for the Steady-State Natural Convection Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {12}, number = {2}, pages = {407--435}, abstract = {

This paper is concerned with two types of two-grid stabilized finite element methods (FEMs) based on Newton iteration for the steady-state nature convection problem. The first method needs to solve one small nonlinear natural convection system on the coarse mesh with mesh width $H$, and then to solve one large linearized natural convection system on the fine mesh with mesh width $h=\mathcal {O}(H^2)$ based on Newton iteration. The other method needs to solve one small nonlinear natural convection system on the same coarse mesh, and then to solve two large linearized systems on the fine mesh with mesh width $h=\mathcal {O}(H^{\frac{7-\varepsilon}{2}})$ based on Newton iteration which have the same stiffness matrix with only different right-hand side. In both methods, the stabilization terms are defined via two local Gauss integrations at element level which has no need to introduce additional variables comparing with the standard variational multiscale stabilized FEMs. The stability estimates and the convergence analysis for both methods are derived strictly. Ample numerical results are presented to confirm the theoretical predictions and demonstrate the efficiency of the new methods.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0156}, url = {http://global-sci.org/intro/article_detail/aamm/13628.html} }
TY - JOUR T1 - Two-Grid Stabilized FEMs Based on Newton Type Linearization for the Steady-State Natural Convection Problem AU - Yang , Yunbo AU - Jiang , Yaolin AU - Kong , Qiongxiang JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 407 EP - 435 PY - 2020 DA - 2020/01 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2018-0156 UR - https://global-sci.org/intro/article_detail/aamm/13628.html KW - Natural convection problem, two-grid method, finite element methods, Newton iteration, variational multiscale stabilization, error estimates. AB -

This paper is concerned with two types of two-grid stabilized finite element methods (FEMs) based on Newton iteration for the steady-state nature convection problem. The first method needs to solve one small nonlinear natural convection system on the coarse mesh with mesh width $H$, and then to solve one large linearized natural convection system on the fine mesh with mesh width $h=\mathcal {O}(H^2)$ based on Newton iteration. The other method needs to solve one small nonlinear natural convection system on the same coarse mesh, and then to solve two large linearized systems on the fine mesh with mesh width $h=\mathcal {O}(H^{\frac{7-\varepsilon}{2}})$ based on Newton iteration which have the same stiffness matrix with only different right-hand side. In both methods, the stabilization terms are defined via two local Gauss integrations at element level which has no need to introduce additional variables comparing with the standard variational multiscale stabilized FEMs. The stability estimates and the convergence analysis for both methods are derived strictly. Ample numerical results are presented to confirm the theoretical predictions and demonstrate the efficiency of the new methods.

Yunbo Yang, Yaolin Jiang & Qiongxiang Kong. (2020). Two-Grid Stabilized FEMs Based on Newton Type Linearization for the Steady-State Natural Convection Problem. Advances in Applied Mathematics and Mechanics. 12 (2). 407-435. doi:10.4208/aamm.OA-2018-0156
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