Requirements of the Numerical Method

In this section we want to consider the connection between the used discretization scheme and the grid quality criteria we have to use.

Finite Elements versus Finite Volume

Let's consider the simplest case of the diffusion equation --- the Laplace equation --- which describes the stationary state of a single species with constant diffusion coefficient: \Delta u = 0 Let's compare now the discretization of this standard example operator for the two most popular discretization schemes --- the Finite Element Method (FEM) and the Finite Volume (FV) discretization. We consider only one question - the M-matrix property for the resulting matrix.

The M-matrix property depends on the sign of the so-called connection value between two neighbour nodes i and j. For FEM, this connection value is defined by: \Gamma_{ij} = - \int \nabla \psi_i \nabla \psi_j dV Here \psi_i is the basic function for the i-th node of the discretization.

In the FV method, for every grid node a box around the node has to be defined. For the most usual FV variant the Voronoi boxes are used. We obtain the following connection value: \Gamma_{ij} = A_{ij} / l_{ij} Here A_{ij} is the area of the common boundary of the Voronoi boxes of the nodes i and j, and l_{ij} the distance between these nodes. In above cases, the (weak) M-matrix property will be fulfilled if \Gamma_{ij} >= 0 for every pair of neighbour nodes i and j.

In 2D, we have a satisfactory situation. The above coefficients coincide, and we have the following well-known For a given set of points, there is only one grid --- the Delaunay grid --- so that for every inner connection we have \Gamma_{ij} >= 0 as for FEM, as for FV.

It is well known that the generalization of this theorem to 3D fails. The Delaunay grid is not the optimal grid for FEM. Lo [Lo1991] thinks that this is a fundamental weakness of 3D Delaunay triangulation. The typical situation is the so-called sliver --- a badly distorted tetrahedron with well-proportioned faces but with arbitrary small volume. Sliver can account for as much as 10 % of the total number of tetrahedra generation. They lead to arbitrary big terms with incorrect sign of the related connection value \Gamma_{ij}.

Sometimes sliver can be removed by a local grid transformation. But, in general, this does not lead to the M-matrix property for the grid. Letniowski [Letniowski1992] has given a simple example of a point set so that every grid has an inner connection with \Gamma_{ij} < 0.

Our opinion is slightly different from the point of view of Lo. If his conclusion is "How bad for the Delaunay grid", we say "How bad for the FEM method". The reason is that the generalization fails only for FEM, not for FV: For the FV method, for a given set of points, there is only one grid --- the Delaunay grid --- so that for every inner connection we have \Gamma_{ij} >= 0.

The proof is a trivial consequence of the formula we have given before for the FV method.

Remark that for the time-derivative part of the diffusion equation\partial_t u very often "mass matrix lumping" will be used because of the numerical problems of the pure FEM formula. This modification also may be considered as going from FEM to some FV method. There seems to be no reason not to do the same for the Laplace term.

Non-Constant Diffusion Coefficient

Consider now shortly how we have to handle a non-constant diffusion coefficient.

One, very popular point of view is to consider the diffusion coefficient as constant over each element. But this strategy can easily lead to an incorrect sign for a connection coefficient. Remark that the connection coefficient \Gamma_{ij} may be described as the sum over all elements containing these two nodes. The previous theorem guarantees that this sum is greater zero for the Delaunay grid, but not that every part is greater zero. If these parts will be weighted in another way by the different diffusion coefficients of the elements, we can obtain resulting coefficients with incorrect sign.

If we consider the diffusion coefficient as constant over the "domain of influence" of the edges (consisting of the two cones over the common boundary of the Voronoi cells), we can avoid this effect. In this case, we have to compute the diffusion coefficient over this domain and to multiply it with the coefficient \Gamma_{ij}.

Boundary Consideration

For boundary edges the previous theorems do not lead to a correct sign of the connection coefficient \Gamma_{ij}. Here we have to modify the point set itself to obtain a grid which leads to an M-matrix property.

The standard technique in this case is to include new boundary points. Usually this technique allows to avoid connection values with incorrect sign. In the case of anisotropic grids, this may be not the ideal solution, and other techniques have to be used to obtain this result.

If a point set fulfills the M-matrix condition also for the internal boundaries, also another problem of Delaunay grids will be solved. In general, the Delaunay grid may not preserve internal boundaries. There may be Delaunay edges between internal nodes of different regions. But we have the following If the grid consisting of different sub-grids for different regions leads to an M-matrix property for the FV method separately for each sub-grid, it will be the Delaunay grid of this point set. Especially, in this case the Delaunay grid preserves the inner boundaries.

Conclusion

We see that we can the M-matrix property by using the following strategy: One question is if the M-matrix property is really so important. An M-matrix is desirable for iterative sparse matrix methods. But it is also possible to solve problems without this property. The main reason for us is that with an M-matrix the approximation satisfies a discrete maximum principle. This leads to nonnegative concentrations if the initial conditions and boundary conditions are correct. Remark, that this is very important especially for diffusion equations, because negative concentrations are physically nonsense. For other applications of the Laplace operator the importance may be not so big. So, for other applications higher accuracy combined with some small negative values may be the better solution.