Decoupled two‐grid finite element method for the time‐dependent natural convection problem I: Spatial discretization

Abstract

In this article, a decoupled two grid finite element method (FEM) is proposed and analyzed for the nonsteady natural convection problem using the coarse grid numerical solutions to decouple the nonlinear coupled terms, and the corresponding optimal error estimates are derived. Compared with the standard Galerkin FEM and the usual two-grid FEM, our algorithm not only keeps good accuracy but also saves a lot of computational cost. Some numerical examples are provided to verify the performances of the decoupled two-grid FEM. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the decoupled two-grid FEM for the nonsteady natural convection problem.

One‐level and two‐level finite element methods for the Boussinesq problem

“…The estimates (4) and (5) indicate that the two-level method shows the same order of approximation as the standard Galerkin FEM under the condition h = (H 2 ). However, in two-level method, the nonlinearities are only treated on the coarse mesh and two small linear problems need to be solved on the fine mesh.…”

“…How to obtain the optimal L 2 ‐norm error estimates for velocity and temperature have bothered the researchers for a long time. Recently, the decoupled schemes are considered for the Boussinesq problem in Zhang and Tao and Zhang et al() In these papers, under some restrictions about the parameters, optimal L 2 ‐norm estimates for velocity and temperature were established for problem in steady case. Furthermore, they established the optimal error estimates of velocity and temperature in different norms for problem in both time and spatial discrete schemes.…”

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

“…Among these numerical schemes, the Crank‐Nicolson extrapolation scheme is almost unconditional stability, while Crank‐Nicolson/Adams‐Bashforth scheme requires some restrictions on the time step and mesh size. Therefore, in this paper, we consider the Crank‐Nicolson extrapolation scheme for the natural convection problem, our work is extension and supplement the previous works and provide some new stability and convergence results for the numerical solutions. At the same time, based on He, He and Li, and He et al, for the 3D Navier‐Stokes equations, we also consider the Crank‐Nicolson extrapolation scheme for 3D time‐dependent natural convection problem.…”

The Crank-Nicolson/Explicit Scheme for the Natural Convection Equations with Nonsmooth Initial Data

“…In this way, the origin problem is split into two linear subproblems, and these subproblems with the constant coefficient matrix can be solved easily in each time level. Compared with [32,33], the main contributions can be list as follows:…”