In this section the principle of the finite element method is demonstrated
by solving the Poisson equation (4.11) in the two-dimensional
domain
. One part of the domain boundary
represents
the Dirichlet boundary
and the remaining part --
the Neumann boundary
, respectively
(
).
As discussed in Chapter 3 the unknown function
is approximated by
as in (3.5).
The known basis functions will be notated with
instead of
.
The reason for this will became clear in the vector finite element chapter,
where systems of partial differential equations appear, which require
a combination of vector and scalar interpolation functions.
The function
complies with the Dirichlet boundary function on
. The set of known functions
builds
a fundamental function system vanishing on the Dirichlet boundary
. To find the numerically approximated solution of
(4.11) for
it is necessary to determine the
unknown multiplier coefficients
. This is performed by substituting
in (4.11) by its approximation (4.13)
and then weighting the resulting residual of (4.11) with the set
of functions
in the domain
Equation (4.14), using the first scalar theorem of Green (3.15) analogously to (3.16), leads to
![]() |
(4.15) |
with the fact that all functions
vanish on
to
The Neumann boundary condition on
can be written
analogously to (3.18) as
using
and
.
After substitution of
in (4.16) by
the approximation (4.13) and using
the Neumann boundary condition (4.17)
the following linear equation system for the unknown coefficients
is obtained
![]() |
(4.18) |
The above equation can be converted to
![]() |
(4.19) |
which complies with the following linear equation system
The matrix
and the right hand side vector
are given by the expressions
Let assume that
is a part of the outer boundary of the two-dimensional
domain
. Furthermore
is assumed sufficiently large to allow that
can be neglected on
. Since the corresponding simulations are
performed in finite domains, this assumption will lead to systematic error which
became smaller with increasing domain size. This issue will be discussed in
Subsection 4.1.5. It is also assumed that there is
no electric charge density distribution
in the domain of interest.
Thus the Neumann boundary term and the source term in (4.21)
are set to zero. Otherwise, if
is perpendicular to
on
, the boundary condition on
will be zero
independently from the domain size.
The function
, which satisfies the Dirichlet boundary
can be analogously to the solution approximation (4.13)
written as a sum of known functions multiplied with coefficients
Now the coefficients
in (4.22) where
are known
values. They are obtained easily from
for
points on the Dirichlet boundary
by the following linear equation system
With (4.22) the right hand side vector
from (4.21) can be rewritten as
In general the form functions
can be defined in the entire simulation domain.
For example, this is the case by the weighted residual method. In the case of the finite
element method the domain
is divided into smaller
sub-domains
. This process is known as domain discretization and
the resulting sub-domains
are called elements. The index
or
will be used in this work generally for quantities, which comply with an element.
It can be considered as a numbering index. The shape
functions
are non-zero only in a few neighboring
elements and vanish in the remaining simulation area.
is a global shape
function. Its local representation in each element, in which it is non-zero
is termed as element shape function
. Now
is written as a superscript index,
because there is already another numbering index available -- the subscript index
.
In each element usually many
element shape functions are defined, which are part of different global shape functions.
As it will be shown later for this purpose low order polynomials are used.
It is very important to notice the difference in the indexing (index
) between
the global and the element basis functions. In the global basis function
complies with the number of global functions in the whole region and with the number of
unknown coefficients
, respectively. By the element basis function
the index
corresponds to the number of the element basis functions in the element.
In practice at first the element form functions
are defined for each element in the domain and then the global ones are constructed by them.
In this work the area of interest is discretized on triangular elements. On these elements linear triangular element functions are employed.