The Finite Element Method on Quadrilateral Meshes

摘要

The theoretical analysis of the finite element method is well established in the case of triangular or tetrahedral meshes. In this case optimal approximation properties have been proved in all reasonable functional norms. Several commercial codes use this method which is now in the common practice of engineering applications. On the other hand, the case of quadrilateral or hexahedral meshes, even if commonly used in the applications (sometimes it seems to be even more popular than the previous one), has not been studied in such deep detail, probably because it hides some insidious issues. For simplicity, we restrict our analysis to the two-dimensional case; three-dimensional analysis in some cases is a straightforward extension, in some others is more complicated. Indeed, we shall show that some commonly used quadrilateral finite elements present a lack of convergence when general regular meshes are used. The list of suboptimal elements includes serendipity (or trunk) elements, face elements, edge elements. A rigorous theory is presented, which gives necessary and sufficient conditions for optimal order approximation. Our theory is supported by several numerical experiments, which are taken from various engineering applications, ranging from elasticity and fluid dynamics to acoustics and electromagnetics. Example of suboptimal elements include: 8-node element for Poisson problem, Q{sub}2 -^sP{sub}1 Stokes element, face elements for acoustic problem, edge elements for electromagnetic problems.