Nonconforming Finite Element Methods for Stokes and Elliptic Problems

Abstract

A new class of nonparametric nonconforming quadrilateral finite elements is introduced in Chapter 1, which has the midpoint continuity and the mean value continuity at the interfaces of elements simultaneously as the rectangular DSSY element [8]. The parametric DSSY element for general quadrilaterals requires five degrees of freedom to have an optimal order of convergence[4], while the new nonparametric DSSY elements require only four degrees of freedom. The design of new elements is based on the decomposition of a bilinear transform into a simple bilinear map followed by a suitable affine map. Numerical results are presented to compare the new elements with the parametric DSSY element.

In Chapter 2, as in two dimension, a class of onparametric DSSY element in three dimensional hexahedral mesh is designed. It satisfies the optimal convergence property in genuine hexahedral mesh (consisting of six flat faces) with six local basis functions contrast to the parametric DSSY element. Also, the case of hexahedron mesh with non-flat faces is discussed. The numerical results for the genuine hexahedral mesh and the non-flat hexahedral mesh are presented.

A new nonparametric nonconforming quadrilateral finite element is introduced in Chapter 3. The finite element is based on the nonparametric DSSY element introduced in Chapter 1 and constructed to have the minimal value of H1-seminorm on each element as in [11]. From this, the finite element has the maximal inf-sup constant and performs better than the nonparametric DSSY element introduced in Chapter 1 in the aspect of computing time. Numerical results are presented to show the speedup of computation by comparing with [13].