The Support-Operators Method

We next describe the support-operators method. It is convenient at this point to define a flux operator given by - D$\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$}$ . The diffusion operator of interest is given by the product of the divergence operator and the flux operator: - $\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$} \cdot$}$D$\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$}$ . The support-operator method is based upon the following three facts:

• Given appropriately defined scalar and vector inner products, the divergence and flux operators are adjoint to one another.

• The adjoint of an operator varies with the definition of its associated inner products, but is unique for fixed inner products.

• The product of an operator and its adjoint is a self-adjoint positive-definite operator.

The adjoint relationship between the flux and divergence operators is embodied in the following integral identity:

$$\oint_{{\partial V}}^{}$$$$\phi$$$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{H}$}$$ ^. $$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{n}$}$$  dA - $$\int_{V}^{}$$D^-1$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{H}$}$$ ^. D$$\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$}$$$$\phi$$  dV =

$$\int_{V}^{}$$$$\phi$$$$\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$} \cdot$}$$$$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{H}$}$$  dV   ,

where $\phi$ is an arbitrary scalar function, $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{H}$}$ is an arbitrary vector function, V denotes a volume, $\partial$V denotes its surface, and $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{n}$}$ denotes the outward-directed unit normal associated with that surface.

Our support-operator method can be described in the simplest terms as follows:

1. Define discrete scalar and vector spaces to be used in a discretization of the integral identity.

2. Fully discretize all but the flux operator in the identity over a single arbitrary cell. The flux operator is left in the general form of a discrete vector as defined in Step 1.

3. Solve for the discrete flux operator (i.e., for its vector components) on a single arbitrary cell by requiring that the discrete version of the identity hold for all elements of the discrete scalar and vector spaces defined in Step 1.

4. Obtain the interior-mesh discretization of the identity by connecting adjacent mesh cells in such a way as to ensure that the identity is satisfied over the whole grid. This simply amounts to enforcing continuity of intensity and normal-component flux at the cell interfaces.

5. Change the flux operator at those cell faces on the exterior mesh boundary so as to satisfy the appropriate boundary conditions.

6. Combine the global divergence matrix and the global flux matrix to obtain the global diffusion matrix.