| In this paper,for the nonlinear convection-diffusion equation with a gener-al capacity term,the basic properties of the fully implicit finite element discrete schemes and their iterative accelerating methods are studied,in order to solve the problem quickly and accurately.Starting from one-dimensional problem,accord-ing to whether the convection is dominant,two cases are studied.In the case of diffusion domination,the full implicit standard finite element method is used to discretize the solution,and in the case of convection domination,the full implicit characteristic modification finite element method is designed to avoid non-physical numerical oscillation and numerical dispersion.In order to solve the nonlinear prob-lem efficiently,the Picard-Newton iterative acceleration method is designed by using the "linearization-discretization" technique.By introducing finite element projec-tion and developing new reasoning techniques,the basic properties of the discrete schemes and iterative methods are strictly analyzed theoretically.It is proved that the solution of the standard nonlinear finite element scheme is uniquely existed,absolutely stable and has the 1st order time and optimal order space L?(L2)con-vergence,the Picard-Newton iteration method has the same convergence precision,and the Picard iteration and the Newton iteration have linear and quadratic con-vergence rates respectively.Moreover,it is proved that the solution of the nonlinear characteristic finite element scheme is existed has the 1st order time and optimal order space L?(L2)convergence.Its Picard-Newton iteration method has the same convergence precision,and the Picard and Newton iteration have linear and super-linear convergence ratios,respectively.Numerical experiments are carried out on uniform grid to verify the theoretical results.The ideas and methods can be ex-tended to multi-dimensional case and second-order time accuracy schemes. |
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