One typical application of the scalar finite element method is the numerical solution of a Poisson equation. Thus it is used for a detailed explanation of the basic concept of this method.
The Poisson equation is derived from the Maxwell equations [41,42] for the static case -- the field quantities do not vary with time. The differential form of the four Maxwell equations is usually given as
where each variable has the following meaning and unit:
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In this work
and
will be also referred to as
electric field and magnetic field, respectively.
Applying the divergence operator to (4.3) and
substitution by (4.4) give
The macroscopic properties of the medium are described in terms
of permittivity
, permeability
, and conductivity
. These parameters are used for specifying the
constitutive relations between the field quantities
In general it is not necessary that the constitutive parameters
,
, and
are simple constants. For example the relationship
between
and
in
(4.7) may by highly non-linear
for ferromagnetic materials. In this case
depends on the
field. For anisotropic media the directions of the flux densities
differ from the directions of the corresponding field intensities
and the constitutive parameters must be described by tensors. In
inhomogeneous regions
,
, and
are functions
of position.
If the field values are invariant in time, the field is static. In this case the magnetic field and the electric field do not interact. For instance, the electrostatic case is given by (4.4) and
Equation (4.9) is satisfied by
After substituting (4.10) in (4.6)
and insertion in (4.4),
the following second order partial differential equation for the electrostatic
potential
is obtained
If assumed that
is a constant scalar,
the expression (4.11) turns into the well known
Poisson equation. If the charge density is zero allover the domain,
(4.11) leads to
which corresponds for constant scalar permittivity
to the Laplace equation.
The most general presentation for the permittivity,
which can be handled by the finite element method in the frequency domain is
a position dependent tensor
.