The Discontinuous Galerkin Finite Element Method For The Stokes Eigenvalue Problem

The generalized Stokes eigenvalue problems（including the classi-cal Stokes eigenvalue problem and the magnetohydrodynamic Stokes eigenvalue problem）play an important role in the stability analysis of the fluid mechanics.We study the finite element method for the problems in this paper,including discontinuous Galerkin finite element method,local discontinuous Galerkin finite element method,noncon-forming Crouzeix-Raviart element and the enriched Crouzeix-Raviart finite element method.The main contents are as follows:Firstly,for the generalized Stokes eigenvalue problems on R^d（d=2,3）,the mixed discontinuous Galerkin finite element method using P_k-P_k-1（k≥1）element is studied in this paper.A complete prior error estimate,the residual posterior error estimate and the a pos-terior error estimator of the approximate eigenpairs are given.The reliability and efficiency of the posteriori error estimator for the eigen-function are proved by using the enrichment operator and the lifting operator and the reliability of the estimator for the eigenvalue is also analyzed.The numerical results are provided to confirm the theoretical predictions and indicate that the discontinuous Galerkin finite element method can reach the optimal convergence order O(dof^-2k/d).The in-fluence of the Hartmann number on the eigenpair for the MHD Stokes eigenvalue problem and the variation of the corresponding eigenstruc-ture with the magnetic field are discussed.The numerical experiments show that the eigenvalue increases with the increase of the Hartman-n number and the magnetic field has a great influence on the flow field.Secondly,a multigrid discretization of discontinuous Galerkin method using P_k-P_k-1（k≥1）element is proposed and its the a priori er-ror estimate are proved in this paper,the posterior error estimator of the approximate eigenpair is given,the reliability and efficiency of the posteriori error estimator for the eigenfunction are proved and the reliability of the estimator for the eigenvalue is analyzed.The numerical results confirm the theoretical analysis and show that the proposed method is effective and can reach the optimal convergence order O(dof^-2k/d).Thirdly,the local discontinuous Galerkin finite element method for the generalized Stokes eigenvalue problems is studied in this pa-per,two two-grid discrete schemes of the local discontinuous Galerkin finite element method are established,the priori error estimate is giv-en and numerical experiments are performed.The numerical results show that the local discontinuous Galerkin finite element method and two two-grid discrete schemes are effective for the generalized Stokes eigenvalue problems.Finally,the residual posterior error estimate and adaptive al-gorithm of nonconforming Crouzeix-Raviart finite element and en-riched Crouzeix-Raviart finite element for the classical Stokes eigen-value problem on R^d（d=2,3）are studied and the a posterior error estimator is given and proved to be reliable and effective.Based on the posterior error estimator,two adaptive algorithms,namely direct solution adaptive algorithm and shift inverse iteration adaptive algo-rithm,are established in this paper.Numerical experiments and the-oretical analysis show that the approximate eigenvalues approach the exact eigenvalues from below and reach the optimal convergence order O(dof^-2/d).