One has

Denoting
the jacobian

one finally has

When
, the expression of the jacobian reduces to
.
With
a part of the boundary of
a real element and
the corresponding boundary on the reference element
,
one has

where
is the unit normal to
on
. In a same
way

For
the unit normal to
on
.
Denoting
![\nabla^2 f =
\left[\frac{\partial^2 f}{\partial x_i \partial x_j}\right]_{ij},](../_images/math/922650db95a714a428af3c384a4e359d258de17e.png)
the
matrix and

the
matrix, then

and thus

In order to have uniform methods for the computation of elementary matrices, the
Hessian is computed as a column vector
whose components are
. Then, with
the
matrix defined as
![\left[B_2(\widehat{x})\right]_{ij} =
\sum_{k = 0}^{N-1}
\frac{\partial^2 \tau_k(\widehat{x})}{\partial \widehat{x}_{i / P} \partial \widehat{x}_{i\mbox{ mod }P}}
B_{kj}(\widehat{x}),](../_images/math/af0dc2d4ddff4d7e7c23ea66da969f7cd3547afb.png)
and
the
matrix defined as
![\left[B_3(\widehat{x})\right]_{ij} =
B_{i / N, j / P}(\widehat{x}) B_{i\mbox{ mod }N, j\mbox{ mod }P}(\widehat{x}),](../_images/math/a154bfb20b9d7666d4d1ac1dc4f0ea257870ff0d.png)
one has

Assume one needs to compute the elementary “matrix”:

The computations to be made on the reference elements are

and

Those two tensor can be computed once on the whole reference element if the
geometric transformation is linear (because
is constant). If the
geometric transformation is non-linear, what has to be stored is the value on
each integration point. To compute the integral on the real element a certain
number of reductions have to be made:
) nothing.
) a
reduction with respect to
with the matrix
.
)` a reduction of
with respect to
with the matrix
and a reduction of
with respect also
to
with the matrix 
The reductions are to be made on each integration point if the geometric transformation is non-linear. Once those reductions are done, an addition of all the tensor resulting of those reductions is made (with a factor equal to the load of each integration point if the geometric transformation is non-linear).
If the finite element is non-
-equivalent, a supplementary reduction of the
resulting tensor with the matrix
has to be made.