On the Compactness of Quasi-Conforming Element Spaces and the Convergence of Quasi-Conforming Element Method

In this paper,the compactness of quasi-conforming clement spices and the convergence of quasi-conforming element method are discussed.The well-known rellich compactness theorem is generalized to the sequences of quasi-conforming element spaces with certain properties,and the generalized poincare inequality,the generalized Friedrichs inequality and the generalzed inequality of Poincare-Friedrichs are proved true for them.The error estimates are also given.It is shown that the quasi-conforming element method is convergent if the quasi-conforming element spaces have the approximability and the strong continuity,and satisfy the rank condition of element and pass test IPT.As practical examples,6-parameter,9-parameter,12-parameter,15-parameter,18-parameter and 21-parameter quasi-conforming elements are shown to be convergent,and tlieir.L^2,2(Ω)-errors are O(h_τ)、O(h_τ)、O(h_τ²)、O(h_τ²)、O(h_τ³)and O(h_τ⁴)respectively.