Finite element analysis for the 2D/3D unsteady incompressible Darcy–Brinkman equations with double‐diffusive convection based on the penalty method

Abstract In this paper, the penalty finite element method for the 2D/3D unsteady incompressible Darcy–Brinkman equations with double‐diffusive convection (IDBDDC) is considered. The introduction of penalty term could overcome the divergence‐free restriction about the velocity field, allowing only a set of decoupled elliptic equations to be solved. Firstly, some a priori regularity estimations of the weak solutions to the penalized IDBDDC are derived, and then the optimal error estimation for the penalized IDBDDC is also derived. Furthermore, the Euler semi‐implicit method is proposed as a time‐stepping technique for the penalized IDBDDC. Based on the above regularity results, the unconditional stability and optimal error estimates for the Euler semi‐implicit method are established. Finally, the effectiveness of the proposed method is demonstrated by several numerical experiments.