A Class Set Of Normal Heat Conduction - Convection Problems, Mixed Finite Element Solution

In this paper, a nonlinear Galerkin mixed element method, a Galerkin/Petrov-least squares-type mixed finite element method and a nonlinear Galerkin/Petrov-least squares mixed element method for the stationary conduction-convection problems are presented and analyzed, respectively. We discuss the existence, uniqueness and convergence. The nonlinear Galerkin mixed element method and the nonlinear Galerkin/Petrov-least squares mixed element method are based respectively on finite element spaces XH and WH defined on a coarse grid with grid size H and on finite element spaces Xh\Xh and Wh\Wh defined on another fine grid with grid size h < H for the approximations of the velocity and temperature, and a finite element space Qh for the approximation of the pressure, defined on fine grid with grid size h. The Galerkin/Petrov-least squares-type mixed finite element method and the nonlinear Galerkin/Petrov-least squares mixed element method so that they are stable for any combination of discrete finite element spaces without requiring the Babuska-Brezzi stability condition.