A Low Order N-dimensional Rectangular Finite Element For Fourth Order Elliptic Problem

The finite element method is one of the main methods to solve elliptic boundary value problems by numerical method and it has been widely used in various engineering areas with the characteristics of its flexibility,rapidity and effectiveness.Nonconforming finite element method has attracted the attention of scientists and engineers in recent years because it can solve partial differential equations with low complexity.From the practical point of view,the geometric shape of finite element division should be as simple as possible.In this paper,the finite element of rectangular subdivision is considered.In this paper,a new rectangular finite element is constructed to solve the fourth order ellipse problem in n-dimensional space.First divide the rectangle into 2n small rectangles,which serve as a macro-element.The basis function is piecewise and C1 continuous on macro-element.The constructed finite element space is C0 continuous on whole region,but not C1 continuous.It is nonconforming for fourth-order problems.The new finite element is(?)(h)convergent in energy norm,and(?)((h2)convergent in H1 norm.The convergence of the finite element method for biharmonic problem is proved strictly,and numerical experiments are carried out to verify it.