This section gives an overview of the basic concepts for discretization schemes in scientific computing. The most important mechanisms are introduced and are then later converted into applicable and well known discretization schemes, such as the above mentioned finite volumes, finite elements, and finite differences. The purpose of this section is to unify different discretization schemes to a common kernel which is then directly converted to software design concepts. The concepts for discretization are mostly derived from an abstract reference discretization scheme [35], which is not a full discretization scheme by itself. Here only the necessary concepts for time-dependent electromagnetic problems which are discussed in subsequent chapters, are introduced, with some additional concepts from literature [32,24,33,36]
The simulation domain, either a simple space or a more complex
space-time, is discretized by means of two dual oriented cell
complexes for the primary and secondary mesh. The two cell complexes
and therefore the primary and secondary mesh are not required to be
physically or geometrically distinct. The logical separation is
required to distinguish the different cochains/differential forms
only. With dual cell complexes, the attribution of boundary conditions
and the treatment of material discontinuities can be greatly
simplified [36]. The use of non-dual cell
complexes, e.g., operator adjointness, is more complex to obtain. The
primary
-cells are identified by
whereas the
corresponding secondary
-cells are identified by
and the default orientation, as given in Figure
2.1 for three dimensions.
![]() |
Time is also subdivided into two dual cell
complexes. Here the 0-cells are time instants indexed with increasing
time. The time interval between
to
is indexed as
, similar to the formulation used by Yee (see Section
2.4.2). The secondary cells are given by
and depicted in Figure 2.2.
The time-dependent equation
![]() |
(2.2) |
![]() |
(2.3) |
A graphical representation is given in Figure
2.3 where two 0-cells are used for
the corresponding time steps
and
and one 1-cell
for the time interval
. The corresponding quantity
is
attached to the time interval
, whereas the quantity
corresponds to the
cells.
This topological time stepping was proven [35] to be consistent with the conservation of charge as well as to preserve the absence of sources for the Faraday and Amperé laws. Based on this unique topological time stepping and on various discrete constitutive links, a variety of numerical time-stepping procedures, including explicit and implicit ones, can be derived.
Balance or conservation equations can be written in local or global forms [24,33], e.g.:
![]() |
(2.4) |
Discretizing constitutive relations determines the link between various cochains which represents the field quantities approximated by the local constitutive equations. This step of discretization offers a great number of different choices [35].
For most of the finite volume and finite element schemes, a field projection is used by:
In the discrete setting the field functions
and
do not belong to the problem's variable. Instead the
magnetic flux cochain
and the magnetic field cochain
are linked by the relation
.
From the cochain
a field function is derived by a
reconstruction operator:
![]() |
(2.6) |
Next, the local constitutive link, Equation
2.5, is used to derive
. Then, the cochain
has to be obtained by means of
a projection operator
, which produces a cochain for each field
function
:
![]() |
(2.7) |
The discrete constitutive link
is then finally given by:
![]() |
(2.8) |
A natural requirement for the reconstruction and projection operators
is that for each cochain
the following relation holds true
[35]:
![]() |
(2.9) |
To obtain a sparse matrix representation the reconstruction process is usually performed locally, so the value of the reconstructed field function in a particular point depends only on the values of the original cochain on the cells in a sufficiently small neighborhood of the point.
During the last decade a comprehensive framework for the reconstruction and projection operators, called mimetic discretization [63,64,65,66,37,25], was derived. This framework analyzes in detail how these two operators can be used to mimic the analytical and continuous nature of partial differential equation.