Researches On Mixed Finite Element Methods For Two Classes Of Nonlinear Equations

This dissertation includes four parts.In the first part, a new conforming rectangular mixed finite element viriational form is constructed for sine-Gordon equations with the bilinear element and zero order R-T element. By using integral identity techniques, derivative transfer and interpolation operator’s characteristics, we get the superclose properties of the in semi-discrete scheme. At the same time, by virtue of the technique of interpolation post-processing technique, the global superconvergence results are obtained.In the second part, a new nonconforming mixed finite element approximation scheme for nonlinear sine-Gordon equations is proposed. By use of two special properties of this element:(a) the consistency error is of order O(h2) one order higher than its interpolation error O(h), when the exact solution belongs to H3(Ω);(b) the interpolation operator is equivalent to its Ritz-pr ejection operator,the superclose properties and superconvergences of exact solution u in H1-norm and intermediate variable p in L2-norm are deduced for semi-discrete scheme, and the optimal order error estimates are obtained for a two order fully-discrete scheme.In the third part, by use of the given estimates of conforming linear trianglar element, a new H1-Galerkin mixed finite element viriational form is constructed for sine-Gordon equations. With the help of the aforementioned methods and skills, we get the superclose property and superconvergence in both semi-discrete form and fully discrete form.In the last part, a new conforming and nonconforming mixed finite element methods are proposed for Nonlinear pseudo-hyperbolic equations respectively, without using Ritz projection, which is an indispensable tool in the traditional finite element analysise, the superclose and the global superconvergence are derived directly through interpolation operater, mean-value tricle and post-processing technique.