Finite Element And Finite Spectral Element Methods For Coefficient Vector Reduction Based On Proper Orthogonal Decomposition(POD)

This thesis studies the coefficient reduction order finite element method of Proper Orthogonal Decomposition(POD).The POD,widely used in engineering,has been combined with other numerical methods in recent years,including finite difference,finite element,natural boundary element,finite spectral element and other methods,to solve many practical engineering problems.The finite element method and finite spectral element method have high computational accuracy in dealing with large engineering problems.In this thesis,these two methods are combined with POD to further improve the POD method for the purpose of faster construction speed and higher efficiency.The finite element and Crank-Nicolson finite spectral element methods are also used to reduce the order of the coefficient vector based on POD.Firstly,the coefficient vector finite element schemes of hyperbolic equations and two-dimensional unsteady Navier-Stokes equations are constructed.Secondly,POD method is used to obtain the numerical solution after order reduction.Finally,numerical examples are presented for illustration.Through two experiments,examples of hyperbolic equations and square cavity flow experiments,the advantages of the reduced coefficient finite element method based on POD method are demonstrated.This thesis also utilizes the matrix analysis method to attest,simplification of the process of theoretical analysis process.Through the theoretical analysis of the classical scheme and the reduced order scheme,it is proved that the reduced order scheme and the classical scheme have the same basis function and precision.The reduced order format reduces the rounding error accumulation and the running time on the premise of ensuring the accuracy.