Introduction

• The purpose of this paper is to present a local support-operators diffusion discretization for arbitrary 3-D hexahedral meshes.

• We use the standard finite-element definition for hexahedra [1].

• The method that we present is a generalization of a similar scheme for 2-D r - z quadrilateral meshes that was developed by Morel, Roberts, and Shashkov [2].

• We assume a logically-rectangular mesh in our derivation for convenience, but the scheme can also be applied to unstructured meshes.

• The diffusion equation that we seek to solve can be expressed in the following general form:

  $${\frac{{\partial \phi}}{{\partial t}}} \mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$} \cdot$} \mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$} \phi$$

  where t denotes the time variable, $\phi$ denotes a scalar function that we refer to as the intensity, D denotes the diffusion coefficient, and Q denotes the source or driving function. It is sometimes useful to express this equation in terms of a vector function, $\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$ , that we refer to as the flux:
  $$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$$ = - D$$\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$}$$$$\phi$$   .