The Conforming And Nonconforming Finite Element Methods For The Elastic Eigenvalue Problems

The finite element method is a numerical method to solve partial differential equations in the scientific research and engineering design field.When using the finite element methods for approximate computation,one expects to obtain approximate solutions as accurate as possible with less computation costs.The multigrid discretizations and the adaptive scheme are the efficient finite element methods to achieve this goal.With the multigrid scheme,the solution of the eigenvalue problem on a fine mesh is reduced to solving an eigenvalue problem on a much coarser mesh as well as solving a linear algebraic system on the more and more fine mesh,and the resulting solution remains an asymptotically optimal accuracy.Because the main computational work is to solve algebraic equations,the multigrid methods improve the efficiency of solving the eigenvalue problems.Because of the automatical mesh-refinement procedure,the adaptive finite element method has a wide application prospect in practical finite element computation.The elastic eigenvalue problems appear in many physics and engineering applications,wherein,the importance of numerical approximate the elasticity eigenvalue problem lies in the stability of different elastic structures used in practical applications,such as beams,plates and so on,depend on the accurate knowledge of the vibration modes of these structures.The exact value of the linear elastic eigenvalue problem is unknown,therefore,the efficient numerical method is a hot subject.It is studied in this dissertation the conforming and nonconforming finite element methods for the linear elasticity eigenvalue problems and the transmission eigenvalue problem of elastic waves arising from inverse scattering theory,and the multigrid and adaptive schemes.The following are the main contents:For the linear elasticity eigenvalue problem with pure displacement boundary,first,it is established the nonconforming CR finite element multigrid discretization,and proved the scheme is locking-free,highly efficient.Besides,a multigrid correction scheme is presented;Then,based on the nonconforming CR finite element discretization,a posteriori error estimate of residual type is analyzed,the reliability and efficiency of the indicator are proved,and an adaptive algorithm is designed;Finally,the guaranteed lower bound of the eigenvalues are obtained by making full use of Poincare inequality.For the linear elasticity eigenvalue problem with the pure traction boundary,the classical nonconforming CR finite element for solving the problem is unstable.In order to settle this problem,the main work is as follows:The term ∫Ωu·vdx is added to the bilinear form,numerical results show that the CR element is still unstable.Therefore,another two new stabilized methods with two stabilization terms are introduced.Theoretical analysis and numerical experiments show that it is stable to use the CR element with the stabilization terms to solve the linear elastic eigenvalue problem with the pure traction boundary.For the mixed eigenvalue problem coming from the HellingerReissner elasticity,a locking free two grid discretization based on shift iteration is constructed.Numerical experiments test the cases of large parameter λ,and the numerical results show that the method is locking free.For the elastic transmission eigenvalue problem,a posteriori error estimates based on H2-conforming finite element are analyzed.Because the stress tensor σ(u)contains Lame parameters μ and λ,the bubble function technique is applied to complete the proof of the reliability and efficiency of the indicators.Numerical experiments test the cases of some large parameter λ.For the modified transmission eigenvalue problem of elastic waves,the bilinear form of the weak formula is uncoercive.The well-posedness of the modified transmission problem is analyzed,and a prior error estimate of the finite element method is proved by using the AubinNitsche technique.The finite element error analysis for the modified transmission eigenvalue problem is completed by using the BabuskaOsborn theory.Finally,numerical experiments are given.