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In this paper, we consider the Crank-Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank-Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two-grid Crank-Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two-grid Crank-Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H 1 -norm and for pressure in L 2 -norm. However, the two-grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes. KEYWORDSCrank-Nicolson extrapolation scheme, error estimates, stability, the natural convection equations, two-grid method MSC CLASSIFICATION 65N15; 65N30; 76D07Math Meth Appl Sci. 2019;42:6165-6191.wileyonlinelibrary.com/journal/mma

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