Heat Conduction - Convection Problems, Mixed Finite Element Method

In this paper, a Crank-Nicolson mixed element method , a nonlinear Galerkin mixed element method for the non stationary conduction-convection problems time second order accuracy fully discrete formats and a two-level mixed element method with backtracing for the stationary conduction-convection problems are presented and analyed,respectively, an error analysis are provided for the Crank-Nicolson method of time discretization applied to spatially discrete Galerkin mixed element approximations of the nonstationary conduction-convection problems. Second order estimates are proven in time under realistic assumptions about the regularity of the solution. The existence and the convergence of the fully discrete format of nonlinear Galerkin mixed element method with time second order accuracy for the non stationary conduction-convection problems are showen. On the basis of some conclusion, we have proved that the schemes have second-order convergence accuracy for the time discretization, a two-level method for resolving the nonlinearity in finite element approximation of the stationary conduction-convection problems is presented. The two-level method involves solving one small, nonlinear coarse mesh system, one Oseen problem on the fine mesh and one linear correction problem on the coarse mesh. Meanwhile, the existence and the convergence of the approximate solutions are shown.