Resolve 3-D Poisson Equation Of Eigenvalue Problem Using Q^rot₁ Finite Element

Q₁^rot element is a simple nonconforming element.It is raised and analyzed by Rannacher and Turek,and to solve the problem of Stokes.It is defined on rectangle grid,and has simple form and less freedoms.For example it has only six freedoms for the 3-D problem.It is widely used in the practical engineering computing.Some existing theoretical analysis show that resolving the equation of crystal micro-structure using Q₁^roteleme-nt has obtained better results.For second-order elliptic eigenvalue problem,Linqun and Lin-Jiafu expand eigenvalue of poisson equation using Q₁^rotelem-ent and obtain asymptotic error estimate.The computational results demonstrate the finite element asymptotic expansion theory for eigenvalue problem provided in[32],and indicateQ₁^rot nonconforming element approximates the exact eigenvalues from either upper or lower bound.For 3-D case,ShiZhongci and WangJiacheng analyze the convergence property of this nonconforming element using gener-lized patch test and F-E-M test,point out that resolving general second-order elliptic boundary value problems using Q₁^rotelement can get convergence results.The amount of calulation is very cumbersome in 3-D domain.Therefore paper about research of 3-D Q₁^rot element are very few,and there are no according numerical experiments at present.This paper introduced the basic formula of Q₁^rot element, discussed the eigenvalue expansion of poisson equation using Q₁^rot element in a cube domain.The theoretical analysis and numerical results demonstrate that the 3-D nonconforming Q₁^rot element eigenvalue extrapolation formula is valid,which enhance the accuracy of the eigenvalue from second order to the four order.