C~0IPG Efficient Algorithm For Eigenvalue Problems Of Fourth-order Differential Operators

C~0interior penalty Galerkin(C~0IPG)method,developed in recent decade,is a new class of Galerkin methods,which combines continuous Galerkin method,discontinuous Galerkin method and the idea of stable techniques.It is a discontinuous Galerkin method for the fourth order differential operator.Compared with the traditional finite element methods,C~0IPG method employs Lagrange basis function,which is easy to be construct-ed,and can efficiently capture smooth solution using high order Lagrange basis functions.Also,it preserves the positive definiteness of the original problem.And the four order differential operator eigenvalue problems,especially the biharmonic eigenvalue problem and the Helmhotz transmission eigenvalue problem,have attracted the attention of many researchers.Hence,it is valuable that solving fourth order eigenvalue by C~0IPG method.This thesis studies using C~0IPG method to solve the biharmonic eigenvalue problem and the Helmholtz transmission eigenvalue problem and obtains the following results.Firstly,based on the a posteriori error indicator in[17],we further present the a posteriori error indicators of the eigenfunctions and eigenvalues for the biharmonic eigen-value problem and Helmholtz eigenvalue problem,prove the efficiency and reliability of the eigenfucntions and the reliability of the eigenvalues.From these error indicators,we give the adaptive C~0IPG algorithms.Numerical experiments verify our theories.Secondly,we combine the exiting results on C~0IPG method with the previous results on tow-grid method to obtain the C~0IPG two-grid methods,which reduces the solution of the eivenvalue on a fine mesh to the solution of the eigenvalue problem on a coars-er mesh and the solution of a linear algebraic system on the fine mesh.And then we further discuss the C~0IPG multi-grid discretiation scheme based on Rayleigh quotient iterations in an adaptive fashion,which reduces the solution of an original eigenvalue problem to the solution on a coarser mesh and the solution of a series of linear alge-braic equations on a finer and finer meshes.In numerical experiments,we employ the8)-degree(8)=2,3)C~0IPG method to show the efficiency of our schemes.Lastly,we solve the biharmonic eigenvalue problem by a famous finite element called Morley element.Based on the a posterior error indicator of Morley element in[46,65]for the biharmonic equation,we further discuss the a posterior error indicator of Morley element for the biharmonic eigenvalue and establish the Morley multi-grid discretization schemes based on Rayleigh quotient iterations and the inverse iteration with fixed shift in adaptive fashion.We present the examples in two and three dimensions.The numerical results show that our schemes are all efficient.