Characteristic Mixed Finite Element Methods For High Order Equation

Charateristic mixed discontinuous finite element methods for high order equations, KdV , Sobolev and four order equations are discussed in this paper. The main features of these methods are lowering the high order equations with mixed finite element method firstly, and using discontinuous (or continuous ) finite element methods with different time discretization for the lower equations. The properties of the numerical solutions are studied.Mixed finite element method , characteristic method and discontinuous Galerkin methods are combined to one dimensional KdV equations. The characteristic mixed discontinuous finite element scheme and the modified method of characteristics with adjusted advection (MMOCAA) are given. Namely, the characteristic method in time and discontinuous finite element method in space are used. Existence and uniqueness, stability and error estimates of the variational solution are proved.The characteristic continuous mixed finite element method , discretized time derivative with characteristic method and discretized space variable using continuous finite element method, is studied for one dimensional sobolev equation. Related estimations of the finite element solution are proved .For a kind of four order parabolic equations, mixed discontinuous space-time finite element method is discussed. The approach is based on the combination of mixed finite element method and. discontinuous space-time finite element techniques. This method is not only suitable for dealing with high order equations but also fit for discontinuous problems with high accuracy in both space and time. We discussed the stability and convergence of the mixed discontinuous space-time finite element solution.