The purpose of this appendix is to demonstrate that the coefficient matrix for our
support-operators method is SPD (symmetric positive-definite) . This is achieved in
the following manner. First we demonstrate that the
**W**
matrix is SPD.
Next we show that the the coefficient matrix for a single-cell problem with reflective
boundary conditions is SPS (symmetric positive-semidefinite) with a one-dimensional
null space consisting of any set of spatially-constant intensities. At this point the
demonstration becomes perfectly analogous to that given in [1] for the 2-D
case. We conclude the 3-D demonstration by giving a brief description of the final
steps. The full details of these steps are given in [1].

The following mathematical preliminaries are discussed in [7]. A matrix,
**B**
is symmetric if and only if

We begin the overall demonstration by showing that the matrix given in
Eq. (35),
**W**
, is SPD. It suffices to show that its inverse, explicitly
given in Eqs. (26) through (31), is SPD. We begin the construction of
**W**^{-1}
by considering Eq. (25) and the
**S**
-matrices that appear in
it. Each of the
**S**
-matrices is a
3 `x` 3
matrix that is uniquely
associated with a vertex, and each of these matrices operates on a 3-vector composed
of the face-area flux components associated with that vertex. We now re-express these
3 `x` 3
matrices as
6 `x` 6
matrices by having them operate on a vector
composed of all six face-area flux components associated with the cell. For
instance, the matrix
**S**^{LBD}
operates on the following vertex face-area flux vector:

Since

- the matrix,
**A**_{n}**P**_{n}**A**_{n}**P**_{n}, must be SPS for each value of*n*, - an SPS matrix multiplied by a positive scalar remains SPS,
- the diffusion coefficient will always be positive,
- the vertex volumes will be positive with any reasonably well-formed mesh,
- the
**A**-matrices will be invertible with any well-formed mesh, - the
**P**-matrices are not invertible,

The next step in the demonstration is to construct the discrete diffusion equations for a single cell with reflective boundary conditions. We neglect the time-derivative term in Eq. (1) and consider only the diffusion operator. Let us assume a solution vector, , of the form given in Eq. (90). In order to use numeric indices for the coefficient matrix of the single-cell system, we number this vector in the usual manner, i.e.,

The first 6 equations for a single cell are the equations for the face-center intensities. For a reflective boundary condition, these equations simply state that the face-area flux component on each face is zero. However, in analogy with Eqs. (45) through (47), we equivalently require that theThe seventh and last row of

To summarize, the coefficient matrix takes the following block form: where

+ **C** + =

It is easily verified that Substituting from Eq. (112) into Eq. (111), we get Since Eq. (113) reduces to Using Eq. (107), it is easily shown that where Since

Thus

The remainder of the demonstration is identical to that given for the 2-D case in [1]. The final steps can be briefly described as follows:

- Given a multi-cell mesh with
*N*cells, the**C**-matrices for each cell are expanded to operate on the global vector of intensities for the entire mesh. This step is conceptually analogous to the expansion of the**S**^{LBD}matrix given in Eq. (83). Since the**C**-matrices are SPS, their expansions must be SPS. - It is shown that the sum of the expanded
**C**-matrices represents the coefficient matrix for entire mesh with reflective conditions on the outer boundary faces. Since the global coefficient matrix is the sum of SPS matrices, it must be SPS. Furthermore, the null space of the full coefficient matrix must be equal to the intersection of the null spaces of the expanded**C**-matrices. - It is shown that the null space of the full coefficient matrix is spanned by
all vectors of constant intensity. This is the correct result because the analytic
diffusion operator has a null space spanned by all constant intensity functions if
the reflective condition is imposed on the entire outer boundary. The analytic
diffusion operator becomes invertible if the reflective condition is replaced with an
extrapolated boundary condition on any portion of the outer boundary surface.
- Finally, it is shown that if the reflective boundary condition is replaced with
an extrapolated condition on any outer-boundary cell face, the
expanded
**C**-matrix for the cell containing the boundary face has a null space that is disjoint from the null spaces of all the other expanded**C**-matrices. Thus the intersection of the null spaces of all the expanded**C**-matrices is the null set. Since the global coefficient matrix is the sum of the expanded**C**-matrices, and the expanded**C**-matrices are SPS, it follows that the global coefficient matrix is SPD.