Modified Quadrilateral Morley Element Method For A Fourth Elliptic Singular Perturbation Problem

Fourth order elliptic boundary value problems are more focus on one direction.For this kind of problem,conforming finite elements and nonconforming finite elements are two com-monly used methods.Because of piecewise polynomial functions of conforming elements need to meet C1 conditions and the number of unit degrees of freedom is higher,people tend to use nonconforming elements to solve the problem.In fact,some nonconforming finite elements have been constructed and applied in practice.For example,it is known that Morley element is not CO nonconforming element.Therefore,many scholars constantly construct new and high-order nonconforming elements to solve fourth order elliptic boundary value problems.However,many nonconforming finite elements could not solve fourth elliptic singular perturbation problem.This paper proposes a modified quadrilateral Morley element method for a fourth order elliptic singular perturbation problem.For the lower part of the bilinear form,it is replaced by bilinear approximation of finite element functions,but quadrilateral Morley element is used in the higher part.This paper shows that the modified quadrilateral Morley element method converges uniformly with the perturbation parameter ?.Finally,numerical example matches our theoretical results very well.