In the present work we propose and analyze a fully coupled virtual element method of high order for solving the two dimensional nonstationary Boussinesq system in terms of the stream-function and temperature fields. The discretization for the spatial variables is based on the coupling $C^1$- and $C^0$-conforming virtual element approaches, while a backward Euler scheme is employed for the temporal variable. Well-posedness and unconditional stability of the fully-discrete problem is provided. Moreover, error estimates in $H^2$- and $H^1$-norms are derived for the stream-function and temperature, respectively. Finally, a set of benchmark tests are reported to confirm the theoretical error bounds and illustrate the behavior of the fully-discrete scheme.
翻译:在目前的工作中,我们建议和分析一种完全结合的虚拟要素方法,在流功能和温度字段方面解决两维非静止布西内斯克系统。空间变量的离散基于1美元和0美元相匹配的虚拟元件组合法,而对时间变量则采用后向电流法。提供了全分解问题的正确性和无条件稳定性。此外,对流功能和温度分别得出了2美元和1美元的误差估计数。最后,报告了一系列基准测试,以证实理论误差界限和全分解法的行为。