In this work a fractional-step DG-FE method for the time-dependent generalized Boussi-nesq equations is proposed and analysed. The scheme is composed of two steps. In the first step the original problem is reduced into several scalar elliptic equations. An intermediate velocity and temperature are solved simultaneously. Then in the second step, the incompressibility constraint is enforced and velocity is corrected to be discretely divergence free. Moreover, the introduced elliptic term in the correction step enables the imposition of correct Dirichlet boundary conditions at each temporal step, avoiding the artificial boundary layer introduced by classical pressure correction method. DG-FE discretization strategy is utilized, in which the discontinuous Galerkin spacial discretization for flow equations is employed to obtain local mass conservation and traditional finite element spacial discretization is adopted for heat equation to reduce degrees of freedom. By choosing different symmetry and penalty parameters, SIPG-FE and NIPG-FE methods can be utilized. The consistency and stability of both methods are proved. Preliminary error estimates proving the optimal spacial order and suboptimal temporal order are carried out. Based on a different error equation and the preliminary error estimates, the optimal temporal convergence order is obtained. Numerical tests including a benchmark simulating square cavity flow are then presented, to verify the theoretical analysis and validate the method.(c) 2022 Elsevier B.V. All rights reserved.
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