L~2 Superconvergence Analysis Of Rectangular 8-Node Spline Finite Element

The finite element method is a powerful tool for numerical computation and an important method to deal with complex engineering problems.The superconvergence of finite element is one of the important researches of finite element method.At present,the superconvergence of the triangular quadratic element has been proved by the element orthogonal analysis method;and the superconvergence result of the L~2 projection of the normal family of rectangular elements have also been proved.The superconvergence points of its L~2 projection are the product type of the corresponding one-dimensional superconvergence points.The spline finite element method is a new numerical method based on the finite element method combining with the spline method.The spline finite element method takes advantage of the spline functions to satisfy a certain piecewise continuity and high approximation accuracy.The derivatives and integrals of spline basis functions and can be accurately calculated through the B-net method,thereby the calculations of the finite element stiffness matrices can be simplified.With the help of the method which has been used in the superconvergence analysis of the triangular quadratic element and the normal family rectangular element,this thesis studies the superconvergence of the rectangular 8-node spline element with respect to L~2 projection.Set Q as a rectangular domain,and divide ? into uniform rectangular elements,which Th is the set of all vertices,edge midpoints and element centers.Then there is a uniform superconvergence estimate(uh-uI)(z)=O(h~4)and the superconvergence(u-uh)(z)=O(h~4)on the point set Th.Under certain boundary conditions,these properties are valid up to the boundary.We have also used several numerical experiments to verify that the superconvergence can indeed be achieved on the point set T_h.