On the compactness of quasi-conforming element spaces and the convergence of quasi-conforming element method

Abstract

In this paper, the compactness of quasi-conforming element spaces 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 generalized 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 the test IPT. As practical examples, 6-parameter, 9-paramenter, 12-paramenter, 15-parameter, 18-parameter and 21-paramenter quasi-conforming elements are shown to be convergent, and their L^2′2(Ω)-errors are O(h_τ), O(h_τ), O(h ²_τ ), O(h ²_τ ), O(h ^′_τ ), and O(h ⁴_τ ) respectively.