| Plate vibration has been widely used in storage tank and construction industry.More and more scholars pay attention to its vibration and damping plate vibration.The damping plate vibration problem can be described by fractional or integer fourth order partial differential equations,which are difficult to solve accurately,so it has important theoretical and practical significance for the numerical simulation of damping plate vibration problem.In this paper,a series of analyses are carried out by the mixed finite element method.The mixed finite element method can reduce the requirement of the smoothness of the finite element space and simplify the finite element interpolation space,so it is accurate and efficient in solving high order differential equations.This paper discusses the mixed finite element method for two kinds of equations.For the initial boundary value problem of the damping plate vibration of integer order,a hybrid finite element scheme is established by introducing intermediate variables.Then,the existence and uniqueness of semi-discrete and backward Euler fully discrete solutions of the damping plate vibration initial boundary value problem are given.The intermediate variables are approximated by linear elements,and the optimal order error estimation is obtained.Finally,a numerical example is given to verify the accuracy of the hybrid finite element scheme for initial boundary value problem of damping plate vibration and the effectiveness of the method.The influence of damping coefficient on vibration frequency and amplitude of damping plate is simulated.Aiming at the initial boundary value problem of fractional order damping plate vibration,a fully discrete hybrid finite element scheme was established by introducing intermediate variables,and the stability,error and convergence order of the solution of the fully discrete scheme were proved.A numerical example was given to verify the effectiveness of the hybrid finite element method. |
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