| This dissertation includes two parts. In the first part, we investigate some reaction diffusion models in HIV Transmissions which include a susceptible-infective model, a homosexual population model and a heterogeneous population model. Existence theorems of positive steady-state solutions and dynamic behavior of the time-dependent solutions of these models are established by the monotone method. Also discussed is the bifurcation of positive steady-state solutions. In the second part, monotone methods are applied to a finite element approximation problem. A finite element approximation for nonlinear elliptic partial differential systems is developed and the monotone iterative scheme is applied to the approximation problem. By using monotone methods, error estimates and existence theorems for finite element approximation are established. In order to speed up the Computation of the iterative scheme, we propose a parallel algorithm which not only speeds up the Computation of the finite element solution but also keep the monotone properties of the sequence of iterations. Numerical simulations for the susceptible-infective model in 1- and 2-dimension problems are done and compared with the analytic conclusion. |
