An efficient two-step algorithm for steady-state natural convection problem

• Published in: International journal of heat and mass transfer 2016-10, Vol.101, p.387-398
• Main Authors: Wu, Jilian, Huang, Pengzhan, Feng, Xinlong, Liu, Demin
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Convection; Convergence; Equal-order element; Finite element method; Linear systems; Local Gauss integration; Mathematical analysis; Mathematical models; Natural convection problem; Stability and convergence; Two-step algorithm
• ISSN: 0017-9310; 1879-2189
• DOI: 10.1016/j.ijheatmasstransfer.2016.05.061

•For natural convection problem, a new two-step method is proposed.•Computes a lower order predictor and a higher order corrector.•Solving one natural convection problem based on the P1–P1–P1 pair.•Solving linearized equations based on the P2–P2–P2 pair.•Our method costs less time than P2–P2–P2 stabilized method to get the same precision. In this paper, we propose a new highly efficient two-step algorithm based on local Gauss integration for the 2D steady-state natural convection problem. The basic idea of the algorithm is to compute an initial approximation for the velocity, pressure and temperature based on a lowest equal-order finite element pair P1–P1–P1, then to solve a linear system based on a quadratic equal-order finite element pair P2–P2–P2 on the same mesh. Next, we give the corresponding stability and convergence of the algorithm, which show that the new two-step algorithm has the same order convergence rate as the quadratic equal-order stabilized finite element method. Finally, some numerical examples show that the new method is efficient, reliable, has good precision and can save a lot of computational time compared with the quadratic equal-order stabilized method.