Two-grid Methods For The Natural Convection Equations With Non-smooth Initial Data

Nonlinear phenomena are a unique phenomenon in nonlinear systems that reflects the nature of their motion,because people have certain limitations on the understanding of nonlinear phenomena,numerical simulation has become an important method to study the nature and behavior of the solutions of these nonlinear equations.For nonlinear partial differential equations,the smoothness of the initial solution has an important influence on the stability and convergence of long-term solutions.Based on the initial data non-smooth condition,this paper analyzes the stability and convergence of the two-grid finite element method for the natural convection equations.Firstly,consider the two-grid method of first-order backward Euler scheme for natural convection equations under non-smooth initial data.The main idea of this method is to solve the nonlinear natural convection problem on the coarse mesh and solve the linear natural convection problem on the fine mesh.Moreover,the linear problem can be decomposed into two sub-problems: one is the Stokes equation,the other is a linear parabolic problem,and these two sub-problems can be computed in parallel.This paper establishes the stability of the numerical scheme and the optimal error estimate of the numerical solution under a certain time step limit,and gives numerical examples to verify the validity of the numerical scheme.Secondly,the linear Crank-Nicolson scheme finite element method for natural convection equations under non-smooth initial data is explored.The two-grid finite element method and the backward Euler scheme two-grid method under non-smooth initial data conditions are presented.This section uses the second-order Crank-Nicolson scheme,and the nonlinear term is completely linearized by the previous time layer numerical solution.Moreover,this scheme solves the constant coefficient linear problem at each time layer,which improves the computational efficiency.This paper theoretically proves the stability and convergence of the numerical solution,and gives numerical examples to verify the validity of the numerical scheme.Finally,the two-grid Crank-Nicolson extrapolation scheme of the natural convection equations is studied.The linear term adopts the implicit Crank-Nicolson scheme,and the nonlinear term adopts the linear recursive method.Establishing a fully discrete Crank-Nicolson extrapolation scheme and a two-grid Crank-Nicolson extrapolation scheme for the problem under consideration.The stability and convergence analysis of the approximate solution are obtained.Comparing the theoretical results and the computational efficiency,it can be seen that by selecting the appropriate thickness grid ratio,the two-grid method and one grid method have the same convergence order,but the two-grid method only needs to deal with nonlinear problems on the coarse mesh.Finally,numerical examples are given to verify the validity of the numerical scheme.