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

Yihui Han ,  Hongliang Li and Xiaoping Xie

10.4208/nmtma.OA-2019-0069

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

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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.  Wellposedness  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.


  • History

Published online: 2019-06

  • AMS Subject Headings

52B10, 65D18, 68U05, 68U07

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