Bulletin of the
Korean Mathematical Society BKMS

In this paper, the Gauge-Uzawa methods for the Darcy-Brinkman equations driven by temperature and salt concentration \linebreak (DBTC) are proposed. The first order backward difference formula is adopted to approximate the time derivative term, and the linear term is treated implicitly, the nonlinear terms are treated semi-implicit. In each time step, the coupling elliptic problems of velocity, temperature and salt concentration are solved, and then the pressure is solved. The unconditional stability and error estimations of the first order semi-discrete scheme are derived, at the same time, the unconditional stability of the first order fully discrete scheme is obtained. Some numerical experiments verify the theoretical prediction and show the effectiveness of the proposed methods.

Keywords: Darcy-Brinkman equations, Gauge-Uzawa, finite element method, incompressible flow

MSC numbers: Primary 65Mxx, 35Qxx, 76Dxx, 76Mxx

Supported by: Research Fund from the Key Laboratory of Xinjiang Province (No.2022D04014); National Natural Science Foundation of China (No.12061075); Xinjiang Key Laboratory of Applied Mathematics (No.XJDX1401).