New one- and two-level Newton iterative mixed finite element methods for stationary conduction–convection problems

In this article, new one- and two-level Newton iterative mixed finite element methods (MFEM) for two-dimensional stationary conduction–convection problems are presented. In both of the methods, we decouple the conduction–convection problem into two systems: one is for the temperature and the other is for the velocity and pressure. Thus much computational time can be saved. In the new Newton MFEM, we first solve the equation for the temperature, then solve a Navier–Stokes problem by Newton iterative method. While in the two-level method, we solve the decoupled conduction–convection problem as same as in the one-level method using a coarse grid firstly, then seek a fine grid solution by solving a linearized problem on a fine grid. Stability analysis is performed and error estimates of the methods are derived, which show that our methods are stable and have a good precision. Numerical experiments are also given, which confirm the theoretical analysis and demonstrate the efficiency of the new methods.