Volume 12, Issue 4
Robust Globally Divergence-Free Weak Galerkin Finite Element Methods for Unsteady Natural Convection Problems

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 1266-1308.

Published online: 2019-06

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• Abstract

This paper  proposes a class of semi-discrete and fully discrete weak Galerkin  finite element methods for unsteady natural convection problems in  two  and three dimensions. In the space discretization, the methods use piecewise polynomials of degrees $k,$ $k-1,$ and $k$ $(k\geq 1)$ for the  velocity, pressure and temperature approximations in the interior of elements, respectively, and piecewise  polynomials of degree $k$ for the numerical traces of velocity, pressure and temperature on the interfaces of elements. In the temporal discretization of the fully discrete method, the backward Euler difference scheme is adopted. The semi-discrete and fully discrete methods yield globally divergence-free velocity solutions. Well-posedness of the semi-discrete scheme is established and a priori error estimates are derived for both the semi-discrete and fully discrete schemes. Numerical experiments demonstrate the robustness and efficiency of the methods.

• Keywords

Unsteady natural-convection, semi-discrete and fully discrete, weak Galerkin method, globally divergence-free, error estimate.

52B10, 65D18, 68U05, 68U07

lihongliang@lsec.cc.ac.cn (Hongliang Li)

xpxiec@gmail.com (Xiaoping Xie)

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@Article{NMTMA-12-1266, author = {Han , Yihui and Li , Hongliang and Xie , Xiaoping}, title = {Robust Globally Divergence-Free Weak Galerkin Finite Element Methods for Unsteady Natural Convection Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {12}, number = {4}, pages = {1266--1308}, abstract = {

This paper  proposes a class of semi-discrete and fully discrete weak Galerkin  finite element methods for unsteady natural convection problems in  two  and three dimensions. In the space discretization, the methods use piecewise polynomials of degrees $k,$ $k-1,$ and $k$ $(k\geq 1)$ for the  velocity, pressure and temperature approximations in the interior of elements, respectively, and piecewise  polynomials of degree $k$ for the numerical traces of velocity, pressure and temperature on the interfaces of elements. In the temporal discretization of the fully discrete method, the backward Euler difference scheme is adopted. The semi-discrete and fully discrete methods yield globally divergence-free velocity solutions. Well-posedness of the semi-discrete scheme is established and a priori error estimates are derived for both the semi-discrete and fully discrete schemes. Numerical experiments demonstrate the robustness and efficiency of the methods.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0069}, url = {http://global-sci.org/intro/article_detail/nmtma/13224.html} }
TY - JOUR T1 - Robust Globally Divergence-Free Weak Galerkin Finite Element Methods for Unsteady Natural Convection Problems AU - Han , Yihui AU - Li , Hongliang AU - Xie , Xiaoping JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1266 EP - 1308 PY - 2019 DA - 2019/06 SN - 12 DO - http://doi.org/10.4208/nmtma.OA-2019-0069 UR - https://global-sci.org/intro/article_detail/nmtma/13224.html KW - Unsteady natural-convection, semi-discrete and fully discrete, weak Galerkin method, globally divergence-free, error estimate. AB -

This paper  proposes a class of semi-discrete and fully discrete weak Galerkin  finite element methods for unsteady natural convection problems in  two  and three dimensions. In the space discretization, the methods use piecewise polynomials of degrees $k,$ $k-1,$ and $k$ $(k\geq 1)$ for the  velocity, pressure and temperature approximations in the interior of elements, respectively, and piecewise  polynomials of degree $k$ for the numerical traces of velocity, pressure and temperature on the interfaces of elements. In the temporal discretization of the fully discrete method, the backward Euler difference scheme is adopted. The semi-discrete and fully discrete methods yield globally divergence-free velocity solutions. Well-posedness of the semi-discrete scheme is established and a priori error estimates are derived for both the semi-discrete and fully discrete schemes. Numerical experiments demonstrate the robustness and efficiency of the methods.

Yihui Han, Hongliang Li & Xiaoping Xie. (2019). Robust Globally Divergence-Free Weak Galerkin Finite Element Methods for Unsteady Natural Convection Problems. Numerical Mathematics: Theory, Methods and Applications. 12 (4). 1266-1308. doi:10.4208/nmtma.OA-2019-0069
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