H～1-Galerkin Mixed Finite Element Method Of Evolution Equation

In this paper, we focus on the study of the applications of H~1-Galerkin mixed finite element method in evolution equations on anisotropic meshes. At first, we consider the case when the two approximation spaces are noncomforming finite elements, by means of the novel techniques and the typical characteristics of the elements, the same error estimates of convergence as the classical methods are presented without requiring the Ritz projection.Next, we study the case when the approximation spaces are the anisotropic linear triangular element and a new conforming triangular Hermite-type one, by taking advantage of a series of new approaches, the convergence analysis of hyperbolic type integro-differential equations is provided, the error estimates are also obtained .Finally, we discuss that the combination of conforming quadrilateral Q\ element and nonconforming quasi-Wilson element on anisotropic meshes. The error estimates of the energy norm and L~2-norm are obtained, which are the same as those of the traditional methods. Thus the results of this paper extend the applicable scope of the mixed finite element methods.