The Three-dimensional EQ^rot₁ Non-conforming Finite Element

The non-conforming EQ₁^rotelement was invented by Lin Q,Tobiska L,Zhou A in 2001.The type-function of corresponding finite element space keeps continuous under integral meaning when it overs single element boundary.The non-conforming character of the three-dimensional non-conforming EQ₁^rotelement in solving second-order elliptic boundary value problems and second-order elliptic equation problems is determined by the reduced continuity property.In two-dimensional case,Qun Lin and Jiafu Lin have studied the convergence and eigenvalue error expansion for the non-conforming EQ₁^rotelement,and They have proved that the non-conforming EQ₁^rotelement approximates eigenvalues from below on rectangular domains.In addition,in 2008,Yang Yidu et al proved the fact that the non-conforming EQ₁^rotelement approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements.In three-dimensional case,because of the unknown number is 0(h^-3)(h is the diameter of the mesh),even if h is not sufficiently small,the amount of storage and calculation is very tremendous which is difficult to be accomplished in general computers.Say nothing of,the two-grid discretization schemes of finite element is applied to three-dimensional non-conforming EQ₁^rotelement,which is so called"three-dimensional trouble".Therefore, there are very few papers about research of three-dimensional non-conforming EQ₁^rot element and few corresponding numerical experiments at present.However,in this paper,we prove that the compatible term error estimation and L₂-norm error estimation for the three-dimensional non-conforming EQ₁^rotelement on the basis of paper.We prove that the eigenvalue error expansion for the three-dimensional non-conforming EQ₁^rotelement on the basis of paper.And further we easily obtain eigenvalue extrapolation algorithm(a posterior error estimate)for the three-dimensional non-conforming EQ₁^rotelement.We prove that the three-dimensional non-conforming EQ₁^rotelement approximates eigenvalues from below.We apply two-grid discretization schemes to the three-dimensional non-conforming EQ₁^rotelement on the basis of paper,we prove that finite element two-grid discretization schemes for the three-dimensional non-conforming EQ₁^rotelement approximates eigenvalues from below when exact eigenfunction is singular.Moreover,we give all the numerical examples.In addition,although the theoretical proof without being seen,the numerical experiments show that finite element two-grid discretization schemes for the three-dimensional non-conforming EQ₁^rotelement approximates eigenvalues from below when exact eigenfunction is not singular.