Such well pronounced requirements based on a different criterion can be
formulated for a specific application of the finite element method.
The basis is the *maximum principle* which is the most important
property of solutions to convection-diffusion equations.
In its simplest form it states that both the maximum and the minimum
concentrations occur on the boundary or at the initial time. This implies
that if the boundary and initial values are positive, the solution must be
positive everywhere and concentrations may never reach negative values. It
is desirable that the employed discretization also satisfies a maximum
principle. As is well known, this is guaranteed, if the system matrix
resulting from the discretization is an M-matrix^{3.2} [69,125].

The system matrix for a simple diffusion with a standard
Galerkin weighted residual approach, linear elements, and backward Euler
time discretization has the following form

where denote the basis functions and is the area (volume) of element . The in-product has a simple geometrical meaning and leads to an angle criterion for each edge in the mesh, which was recently introduced by [189]. It is an important consideration in three-dimensional finite element mesh generation for diffusion applications with a high concentration gradient.

In two dimensions (3.10) can be written as

and therefore

Hence, multiplying (3.11) with results in

which is equivalent to

Due to (3.12) and (3.15) the finite element mesh criterion (Crit. 3.3) can be expressed in two dimensions as

In two dimensions (3.7) describes the relation
between the circumcircle radius, the edge length, and the opposite angle in
a triangle.
In three dimensions a relation for the circumsphere radius, the edge
length, and the opposite dihedral angle
(which is important for Crit. 3.3) in a tetrahedron does not
exist as was explained in Fig. 3.3.
This leads to a very interesting conclusion.
It can be shown due to the existing relation (3.7)
that the finite element mesh requirement (Crit. 3.3) is in
two dimensions identical to the box integration mesh requirement
which is based on empty circumcircles (Crit. 3.1 and Delaunay
Crit. 5.2).
To see this equivalence of the angle condition (3.16) and
the Delaunay criterion dependent on (3.7) consider
the extreme case where (3.16) becomes

and furthermore with (3.7)

The two triangles with the common edge (Fig. 3.5) must possess circumcircles with equally sized radii. Because of (3.17) the circumcircles must be in fact identical. Each circumcircle passes therefore through all four vertices of the two triangles and the Delaunay criterion is ``just'' fulfilled. With a decreasing sum the distance between the two circumcenters (centers of the circumcircles) increases and the Delaunay criterion is definitely satisfied.

As expected, in two dimensions the finite volume and the finite element method lead to the same discretization with identical requirements. They both rely on Delaunay meshes to fulfill the maximum principle. For other finite element applications than diffusion, like stationary problems or problems with less high gradients, the use of Delaunay meshes can be omitted. In three dimensions Crit. 3.3 and the Delaunay criterion are of quite different nature as will be shown with simple examples in the next section. In practice finite element mesh generators may generally try to avoid extremely obtuse (dihedral) angles and badly shaped elements without too much concern on the Delaunay property and without a technique to directly enforce Crit. 3.3. Such a technique remains open to further research.

2000-01-20