A new two grid variational multiscale method for steady‐state natural convection problem

Abstract

A two-grid variational multiscale method based on two local Gauss integrations for solving the stationary natural convection problem is presented in this article. A significant feature of the method is that we solve the natural convection problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations firstly, and then find a fine grid solution by solving a linearized problem on a fine grid. In the computation, we introduce two local Gauss integrations as a stabilizing term to replace the projection operator without adding other variables. The stability estimates and convergence analysis of the new method are derived. Ample numerical experiments are performed to validate the theoretical predictions and demonstrate the efficiency of the new method.

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

“…That can save a large amount of the computation time compared with the one‐level method. Two‐level method has also been applied to many other nonlinear equations, such as the Navier‐Stokes equations, the natural convection equations, the reaction diffusion equations, the miscible displacement equations, the Cahn‐Hilliard equations, and the nonlinear parabolic equation . In Huang et al, a novel two‐step method was proposed for the Navier‐Stokes equations, which uses a lower‐order element to solve one nonlinear system as an initial approximation and then uses a higher‐order element to solve one linear system.…”

Error estimates of a two‐grid penalty finite element method for the Smagorinsky model

“…So two-grid method has been massively studied in recent years. For example, we can refer to previous studies [20][21][22][23][24][25][26] for the research of the Navier-Stokes equations and Kong and Yang [27] and Yang et al [28] for natural convection problem. Another main difficulty is that velocity and pressure are coupled, while the penalty method is an effective method to overcome this difficulty.…”