The Interpolation Correction Of The Finite Element Based On Generalized Rayleigh Quotient

Finite Element Methods (FEM) have played an important role in solving numerical differential equations. With the development of electronic computers and various numerical solutions, FEM begins to play an important part in modern structural mechanics, thermodynamics , fluid dynamics, mechanism design, and many other fields. It has become one of the most efficient numerical methods in a wider range. However, how to improve the precision of FEM to solve more complicated problems based on adding no additional computation or adding computation as little as possible, it has been one of the issues of concern. A great deal of papers before discussed how to improve the accuracy of finite element solutions mostly based on selfadjoint eigenvalue problems. Solving selfadjoint eigenvalue problems will not be much more difficult than solving nonselfadjoint counterparts. Therefore, it is necessary to design a highly efficienct finite element scheme for nonselfadjoint eigenvalue problems. This paper discusses high precision methods of the interpolation correction of finite element for elliptic eigenvalue problems, and combines grid local refinement (r-refinement) with the interpolation correction scheme to obtain a higher precision in the reentrant corner. Based on the work of Lin Qun, the interpolation correction and two-grid discretization schemes based on generalized Rayleigh quotient are established to solve nonselfadjoint elliptic problems. We obtain better results of the second-order elliptic problem, which is solved by the interpolation correction of triangular linear element, bi-quadratic element and trilinear element, and present numerical examples which are solved by acceleration technique based on Rayleigh quotient and generalized Rayleigh quotient for selfadjoint and nonselfadjoint elliptic eigenvalue problems. The performance of the approach is illustrated with theoretical analysis and numerical experiments.