Regularity estimations of the 2D/3D unsteady incompressible Darcy-Brinkman equations with double-diffusive convection and their finite element analysis based on incremental pressure correction method

In this paper, the 2D/3D unsteady incompressible Darcy-Brinkman equations with double- diffusive convection are considered. Firstly, several a priori regularity estimates of the weak solutions are derived, and then two fully decoupled incremental pressure correction finite element methods (IPC FEMs) are proposed, i.e., the first-order and second-order standard IPC (SIPC) methods. Based on the above regularity results, the unconditional stability and optimal error estimates for the first-order SIPC method are proved, and then the unconditional stability for the second-order SIPC method is established. Some numerical experiments are carried out to illustrate the effectiveness of the proposed methods.