Lower Spectral Bounds And Multigrid Discretizations For Eigenvalue Problems

In this paper,we explore finite element solutions of eigenvalue problems from two perspectives.on one hand,we discuss lower spectral bounds including asymptotic lower spectral bounds and guaranteed lower spectral bounds.On the other hand,we discuss the multigrid discretizations including the two-grid discretization of Ciarlet-Raviart mixed method,the multigrid correction and adaptive finite element method.On the lower spectral bounds of eigenvalue problems,first of al-l,we discuss asymptotic lower spectral bounds for the second-order elliptic operator with variable coefficients and Stokes operator on d-dimensional domains?d=2,3,···?.Using four types of noncon-forming finite elements including the Crouzeix-Raviart,the enriched Crouzeix-Raviart,the rotated Q₁and the enriched rotated Q₁finite elements,we propose a correction method to nonconforming finite ele-ment eigenvalue approximations and prove that the corrected eigenval-ues converge to the exact ones from below.And the corrected eigenval-ues still maintain same convergence order as uncorrected eigenvalues.The new results remove the constraints of eigenfunction being singular or the coefficients of eigenvalue problems being constants.Secondly,for the Steklov eigenvalue problems with variable coefficients and the Steklov eigenvalue problem in inverse scattering on d-dimensional do-mains?d=2,3?,by executing new correction to the Crouzeix-Raviart and the enriched Crouzeix-Raviart finite element eigenvalue approxi-mations,we obtain similar theoretical results to the second-order el-liptic and Stokes eigenvalue problems.Finally,we obtain guaranteed lower bounds of eigenvalues for two spectral problems arising in flu-id mechanics by using the min-max principles of weak form that is derived by the principles of operator forms.These two problems are the Laplace model for fluid-solid vibrations and the sloshing problem,and all bilinear forms associated with them are positive semi-definite in H¹???.We deal with this difficulty by adding some constraints to solution space and finite element space.On the multigrid discretizations,fist of all,for biharmonic eigen-value problems with clamped boundary condition in Rⁿwhich include plate vibration problem and plate buckling problem,we study the two-grid discretization based on the shifted-inverse iteration of Ciarlet-Raviart mixed method.With our scheme,the solution of a biharmonic eigenvalue problem on a fine finite element space can be reduced to the solution of an eigenvalue problem on a coarser finite element space and the solution of a linear algebraic system on the fine finite element s-pace.With a new argument which is not covered by existing work,we prove that the resulting solution still maintains an asymptotically optimal accuracy when mesh sizes H>h?O?H²?.Secondly,we propose a multigrid correction scheme to solve the Steklov eigenvalue problem in inverse scattering.With this scheme,solving an eigenvalue problem in a fine finite element space is reduced to solve a series of boundary value problems in fine finite element spaces and a series of eigenvalue problems in the coarsest finite element space.We prove error estimates of eigenvalues and eigenfunctions.Finally,we further discuss the a posteriori error estimates and adaptive algorithm for a new Steklov eigenvalue problem,that is the Steklov eigenvalue problem in inverse scattering.We introduce error indicators for primal eigen-function,dual eigenfunction and eigenvalue.And we use G???rding's inequality and duality technique to give upper and lower bounds for the energy norm of error of finite element eigenfunction,which show that our indicators are reliable and efficient.For the above considered eigenvalue problems and proposed meth-ods,we give numerical experiments which coincide in the theoretical results.