If the vector differential operator
from (3.3) is used the corresponding
excitation function in (3.1) must be a vector function as well which
is written as
.
The approximation (3.5) corresponds to
where
is a known function which exactly
fulfills the Dirichlet boundary condition on
.
The basis functions
build a set
of linear independent known functions. Their tangential components vanish
on the Dirichlet boundary
. The residual is given
analogously to (3.7) by
and does not vanish in general. In this case the weighting
functions in Section 3.2 must also be merged into vector
functions
and the dot product must be used instead of scalar
multiplication.
or
With these considerations
the weighted residual method described in section 3.2 will
also lead to the linear equation system (3.12) where the
matrix
and the right hand side
are given by
to the first integral from the left hand side of (3.24), the following is obtained:
Since the tangential component of
on the Dirichlet boundary
is zero the boundary integral of
(3.27) can be written as
Similarly to (3.18) for the scalar function
and its approximation
, the expression
corresponds with the boundary condition
on
[35,39,40]
Consequently for
and
it can be written
where
is given by (3.3).
For both, the scalar case (3.19) and the vector one (3.30),
the matrix
is symmetric.
The previous considerations are based on a three-dimensional domain. For the two-dimensional case similar formulas can be written. This will be explained by an example using Gauß's law:
Let the three-dimensional domain
be a
cylinder with an arbitrary basal plane
(with boundary
) and the height
.
The function
is only defined in the
two-dimensional domain of the cylinder basal plane. Thus
it has no normal component to the cylinder basal
plane and does not depend on the height. Taking these
considerations into account the left and the right
hand side of (3.31) can be
written as
![]() |
which leads to Gauß's law for the two-dimensional case
![]() |
In a similar way the formulas for the
three-dimensional case used in this work can be easily
rewritten for two-dimensional regions.
Thereby the three-dimensional domains
are replaced by two-dimensional ones
. The boundaries
of the three-dimensional regions are replaced
by the boundaries
of the
corresponding two-dimensional ones. The integration
variable must also be changed accordingly.