TABLE OF CONTENTS If the finite element spaces being used to solve an H m-elliptic problem do not lie in the Sobolev space H m( ?), we refer to them as n onconforming elements. In this case, convergence is by no means obvious. Moreover, in addition to the approximation error, there is now an error called the consistency error. To analyze the situation, we need certain generalizations of C a s lemma. We shall apply these to a simple nonconforming element. We also show how they can be used when the conformity fails in a completely different way and the boundary conditions are relaxed.
As usual, let 
. We replace the given variational problem
by a sequence of finite-dimensional problems: Find u h ? S h with
Here the bilinear forms a h are assumed to be uniformly elliptic, i.e., there exists a constant ? > 0 independent of h such that
Our error estimates for nonconforming methods are based on the following generalizations of C a s lemma. For the first generalization, we do not require that a h be defined for all functions in V. In particular, we permit the evaluation of a h using quadrature formulas involving point evaluation functionals which are not defined for H 1 functions. However, we still require that S h ? V.
1.1 First Lemma of Strang. Under the above hypotheses, there exists...
