Some Remarks on Nonconforming Elements and their Applications

In this talk we will discuss some recent developments in nonconforming finite element methods and their applications. In 1973 the linear nonconforming finite elements for triangles or tetrahedrons and a cubic nonconforming element for triangles were developed by Crouzeix and Raviart. Corresponding quadrilateral elements have been proposed by Han (1985), and Rannacher and Turek (1992), Chen (1993), Arbogast and Chen (1995) and later the DSSY nonconforming element introduced by Douglas et al. in 1999, which has been applied to solving Maxwell and Helmholtz equations. Later, Park and Sheen (2002) developed P1 -nonconforming quadrilateral nonconforming elements, which has only 3 degrees of freedom for quadrilaterals instead of 4 degrees of freedom. A quadratic nonconforming element on rectangles has been proposed recently by Lee and Sheen (2006). The incomplete biquadratic element has degrees of freedom similar to those of Morleys element, which consist of values at vertices and normal derivative values at midpoints of edges, while our element has those similar to the element of Fortin and Soulie, which consist of values at two Gauss points of each edge. Several aspects of comparative analyses of the above three elements in two or three dimensional problems will be discussed. Such nonconforming elements have been proved very effectively applicable to mechanics and elasticity.