一个新的非协调四边形膜元

A NEW NONCONFORMING QUADRILATERAL MEMBRANE ELEMENT

利用分析Ｓｐｅｃｈｔ元的技巧，构造了一个新的五节点非协调四边形单元．它对任意四边形网格通过Ｉｒｏｎｓ分片检查和广义分片检查，且单元上的形状函数不依赖于单元本身．同时证明该单元具有一个与Ｗｉｌｓｏｎ元收敛性质相反的特殊性质，即在解适当光滑ｕ∈Ｈ３（Ω）时，其逼近误差为Ｏ（ｈ），相容误差为Ｏ（ｈ２）．这是其它二阶膜元所不具有的．算例表明，该单元有很好的数值效果．这对进一步研究某些超收敛性有重要意义．

A new nonconforming arbitrary quadrilateral element is proposed by using the analysis techniques of Specht element. It is proved that this element pass through Irons′ patch test and the generalized patch test for arbitrary quadrilateral meshes. The choice of shape function is independent on the element′s geometrical parameters. At the same time, it is also proved that this element has a very special property: i.e. the consistency error is one order higher than that of the interpolation error under the assumption of u∈H 3(Ω) . This is precisely contrast to that of the Wilson element and seems never to be found for other membrane finite elements. Numerical results show that this element has good convergence behavior. It is of great importance for superconvergence studying.