The supercloseness property and global superconvergence analysis for the transient conduction-convection equations

Abstract

In this paper, we consider the implicit Euler scheme of the transient conduction-convection equations. The supercloseness properties and global superconvergence results are derived for two pairs of low order rectangular elements. Firstly, we obtain a prior estimate of finite element solutions. Then using the properties of the Stokes projection and Stokes operator, the derivative transforming skill and the $H^{-1}$-norm estimate, we deduce the supercloseness properties of the velocity and temperature in $L^{\infty }\left(H^1\right)$-norm, and the pressure in $L^{\infty }\left(L^2\right)$-norm. Next, the global superconvergence results are obtained through the interpolation postprocessing technique. Finally, a numerical example is provided to confirm the theoretical analysis. Compared with previous results, the superconvergence analysis is more concise, a lower regularity of solutions is required, and no time step restriction is needed.