Researches On Nonconforming Finite Element Methods For The Hyperbolic Equations

This paper mainly includes the following three parts.Firstly, we study two fully-discrete finite element method schemes for dual phase lagging heat conduction equation by nonconforming EQ₁rot element and the zero order Raviart — Thomas （R — T） element. On the one hand, we construct the mixed finite element approximation scheme. On the other hand, throughout bringing in the variable Q= ut, we translate the equation into the ordinary parabolic equations and propose the corresponding Crank — Nicolson fully-discrete scheme. Based on the the special characters of these elements and the high precision analysis, super-close estimates of the original variable and the flux variable are obtained through different skiis.Then, we propose the nonconforming Galerkin finite element approximation by the constrained nonconforming rotated CNR Q₁ element which processing less degrees for the nonlinear hyperbolic equations and deduce the super-close estimates for semi-discrete scheme and a linearized fully-discrete scheme based on the high accuracy analysis and the special character, combining with derivative delivery technique, mean-value skill and a new splitting technique. Meanwhile, we carry out a numerical example to verify the theoretical analysis and the feasibility of the linearized fully-discrete scheme.At last, we study an H¹-Galerkin nonconforming mixed finite element method for pseudo-hyperbolic equations with the CNR Q₁ element and the zero order R—T element. Through the similar methods and skills with the above analysis, the super-close estimates of the original variable u in Hl norm and the intermediate variable p in H（div） norm are obtained respectively for semi-discrete scheme and fully-discrete scheme. Finally, we carry out a numerical example to confirm the theoretical analysis, and show the efficiency of the method.