Error Estimate For A Quadratic Nonconforming Finite Element On Quadrilaterals

The finite element method has wide applications in engineering technology,and also it is an important numerical method for solving differential equations.Nonconforming finite element methods have successfully provided the stable numerical solutions of many practical fluid flow and solid mechanics problems:for instance,linear or nonlinear Stokes problem and the elasticity related problems.Recently nonconforming finite element methods have attracted increasing attention from scientists to engineers in more wide areas.Although the triangular meshes are popular to use,in many cases one wishes to use quadrilateral meshes with appropriate elements instead,when the problem geometry is of quadrilateral nature,especially in three dimensions.This thesis is based on a new quadratic nonconforming finite element on quadrilaterals, which is proposed by Luo and Sheen.The aim of this thesis is using the element solving second-order elliptic problems in two dimensions and obtaining its error estimate.First,we change the boundary value problem of the elliptic equation into its variational problem.Then we use the second Strang lemma estimating the error between the solution of this nonconforming finite element method and the solution of its variational problem,and also the duality argument is used for the error estimate of the two solutions in L~2-norm for the elliptic equation.Error estimates of optimal order are derived in both energy-norm and L~2-norm for the second-order elliptic equation with Dirichlet and Robin boundary value conditions.The errors of the two boundary value problems in energy-norm are both O(h~2),the errors of that in L~2-norm are both O(h~3).