New error analysis of a second order BDF scheme for unsteady natural convection problem

摘要

In this paper, superconvergence analysis of a mixed finite element method (MFEM) is proposed for the unsteady natural convection problem. The system is discretized by the Q_(11)/Q_0 conforming finite elements in space and the linearized second-order BDF method in time. Based on the high accuracy characters of the elements, the superclose and global superconvergence results of order O(h~2 + τ~2) for the velocity u, the temperature T in H~1-norm and the pressure p in L~2-norm are deduced rigorously. Here, h is the subdivision parameter and r, the time step. The main difficulties caused by the complicate convection terms and the couplings between u, p and T are treated with new skillful techniques. Numerical results are provided to confirm the theoretical analysis.