Nonconforming Quadrilateral Finite Element Method For Fourth Order Elliptic Singular Perturbation Problems

The finite element method is a very efficient numerical method for the partial differential equations,which is widely used in scientific computing and engineering fields.The nonconforming finite element methods successfully provide stable numerical solution for many practical fluid flow and solid mechanics problems.Therefore,the nonconforming finite element methods draw wide attention from scientists and engineers in recent years.In this thesis,a nonconforming finite element method for fourth order elliptic singular perturbation problems are discussed.For the reason of technique difficulties,nonconforming elements are always employed to solve this kind of problems,however,not all finite elements are convergent for fourth order elliptic singular perturbation problems.This paper presents a new finite element method to solve a fourth order elliptic singular perturbation problem. The method uses arbitrary quadrilateral meshes and is conforming to the second order elliptic problem, but nonconforming to the fourth order elliptic problem. We proved it is convergent uniformly in the perturbation parameter. Finally, numerical results validate our theoretical results.