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Volume 26, Issue 4
Robust Globally Divergence-Free Weak Galerkin Finite Element Methods for Natural Convection Problems

Yihui Han & Xiaoping Xie

DOI: 10.4208/cicp.OA-2018-0107

Commun. Comput. Phys., 26 (2019), pp. 1039-1070.

Published online: 2019-07

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  • Abstract

This paper proposes and analyzes a class of weak Galerkin (WG) finite element methods for stationary natural convection problems in two and three dimensions. We use piecewise polynomials of degrees k, k−1, and k (k ≥1) for the velocity, pressure, and temperature approximations in the interior of elements, respectively, and piecewise polynomials of degrees l, k, l (l=k−1, k) for the numerical traces of velocity, pressure and temperature on the interfaces of elements. The methods yield globally divergence-free velocity solutions. Well-posedness of the discrete scheme is established, optimal a priori error estimates are derived, and an unconditionally convergent iteration algorithm is presented. Numerical experiments confirm the theoretical results and show the robustness of the methods with respect to Rayleigh number.

  • Keywords

Natural convection, weak Galerkin method, globally divergence-free, error estimate, Rayleigh number.

  • AMS Subject Headings

52B10, 65D18, 68U05, 68U07

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

1335751459@qq.com (Yihui Han)

xpxie@scu.edu.cn (Xiaoping Xie)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-26-1039, author = {Han , Yihui and Xie , Xiaoping}, title = {Robust Globally Divergence-Free Weak Galerkin Finite Element Methods for Natural Convection Problems}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {4}, pages = {1039--1070}, abstract = {

This paper proposes and analyzes a class of weak Galerkin (WG) finite element methods for stationary natural convection problems in two and three dimensions. We use piecewise polynomials of degrees k, k−1, and k (k ≥1) for the velocity, pressure, and temperature approximations in the interior of elements, respectively, and piecewise polynomials of degrees l, k, l (l=k−1, k) for the numerical traces of velocity, pressure and temperature on the interfaces of elements. The methods yield globally divergence-free velocity solutions. Well-posedness of the discrete scheme is established, optimal a priori error estimates are derived, and an unconditionally convergent iteration algorithm is presented. Numerical experiments confirm the theoretical results and show the robustness of the methods with respect to Rayleigh number.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0107}, url = {http://global-sci.org/intro/article_detail/cicp/13228.html} }
TY - JOUR T1 - Robust Globally Divergence-Free Weak Galerkin Finite Element Methods for Natural Convection Problems AU - Han , Yihui AU - Xie , Xiaoping JO - Communications in Computational Physics VL - 4 SP - 1039 EP - 1070 PY - 2019 DA - 2019/07 SN - 26 DO - http://doi.org/10.4208/cicp.OA-2018-0107 UR - https://global-sci.org/intro/article_detail/cicp/13228.html KW - Natural convection, weak Galerkin method, globally divergence-free, error estimate, Rayleigh number. AB -

This paper proposes and analyzes a class of weak Galerkin (WG) finite element methods for stationary natural convection problems in two and three dimensions. We use piecewise polynomials of degrees k, k−1, and k (k ≥1) for the velocity, pressure, and temperature approximations in the interior of elements, respectively, and piecewise polynomials of degrees l, k, l (l=k−1, k) for the numerical traces of velocity, pressure and temperature on the interfaces of elements. The methods yield globally divergence-free velocity solutions. Well-posedness of the discrete scheme is established, optimal a priori error estimates are derived, and an unconditionally convergent iteration algorithm is presented. Numerical experiments confirm the theoretical results and show the robustness of the methods with respect to Rayleigh number.

Yihui Han & Xiaoping Xie. (2019). Robust Globally Divergence-Free Weak Galerkin Finite Element Methods for Natural Convection Problems. Communications in Computational Physics. 26 (4). 1039-1070. doi:10.4208/cicp.OA-2018-0107
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