  It is shown in this paper that non-conforming finite elements on the triangle
using $P^{1}$-nonconforming polynomials and $P^{2}$ -conforming polynomials can
be easily built and used.They appear as an 'enriched' version of the standard
piecewise quadratic six-node element.This work is divided into two parts.In the
first we present the basic- property of the element,namely how it can be built
and basic error estimates.We have observed that this new element behaves like
$P^{1}$ non-conforming element.In the second part we have applied our element
to the elliptic problem and the theoretical estimate has been guaranteed by
numerical result.
