Computational Results

In this section we perform two sets of calculations. The first set demonstrates that our support-operators method converges with second-order accuracy for a problem with a material discontinuity and a non-smooth mesh. The second set demonstrates the effectiveness of our preconditioner as a function of mesh skewness. There are three types of meshes used in all of the calculations: orthogonal, random, and Kershaw-squared. Every mesh geometrically models a unit cube, and the outer surface of each mesh conforms exactly to the outer surface of that cube. Each orthogonal mesh is composed of uniform cubic cells having a characteristic length, l_c . The random meshes represent randomly distorted orthogonal grids. In particular, each vertex on the mesh interior is randomly relocated within a sphere of radius r₀ , where r₀ = 0.25l_c . These random meshes are both non-smooth and skewed, but these properties are approximately constant independent of the mesh size. The Kershaw-squared meshes are a 3-D variation on the 2-D Kershaw meshes that first appeared in [13]. An example of a 20 x 20 x 20 Kershaw-squared mesh is shown in Fig. 7.

Figure 7: A 20 x 20 x 20 Kershaw-squared mesh.

This mesh becomes increasingly non-smooth and skewed as the mesh size is increased.

The problem associated with the first set of calculations can be described as follows:

$${\frac{{\partial \phi}}{{\partial z}}}$$ (70)

for z $\in$ [0, 1] , where

$$\left\{\vphantom{ \begin{array}{ll} D_1 & \hspace{0.25in},\mbox{... ...}, \\ D_2 & \hspace{0.25in},\mbox{for $z \in [0.5,1]$}, \end{array} }\right. \begin{array}{ll} D_1 & \hspace{0.25in},\mbox{for $z \in [0,0.5]$}, \\ D_2 & \hspace{0.25in},\mbox{for $z \in [0.5,1]$}, \end{array}$$ (71)

with a reflective boundary condition at z = 0 , a Marshak vacuum boundary condition at z = 1 , and where D₁ = ${\frac{{1}}{{30}}}$ , D₂ = ${\frac{{1}}{{3}}}$ , and Q = 1 . We refer to this problem as the two-material problem. The exact solution to the two-material problem is:

$$\phi \left\{\vphantom{ \begin{array}{ll} a + b + c_1 z^4 & \hspace{0... ...a + c_2 z^4 & \hspace{0.25in},\mbox{for $z \in [0.5,1.0]$}, \end{array}}\right. \begin{array}{ll} a + b + c_1 z^4 & \hspace{0.25in},\mbox{for $z... ..., \\ a + c_2 z^4 & \hspace{0.25in},\mbox{for $z \in [0.5,1.0]$}, \end{array}$$ (72)

where

$${\frac{{Q(1+8D_2)}}{{12D_2}}} {\frac{{Q\left( D_2 - D_1 \right)}}{{192 D_1 D_2}}} {\frac{{Q}}{{12D_1}}} {\frac{{Q}}{{12D_2}}}$$ (73)

This problem is solved in 3-D on a unit cube having the vacuum boundary condition on one side of the cube together with reflecting conditions on the remaining five sides. We have performed several calculations for the two-material problem with meshes of various sizes. Each calculation uses a mesh with an average cell width that is half that of the preceding calculation. The relative L₂ intensity error was computed for each calculation. This error is defined as the L₂ norm of the difference between the vector of exact cell-center intensities and the vector of computed cell-center intensities divided by the L₂ norm of the vector of exact cell-center intensities, i.e., $\left.\vphantom{\Vert \hat{\phi}_{exact} - \hat{\phi}_{computed} \Vert _2 }\right.$|$\hat{{\phi}}_{{exact}}^{}$ - $\hat{{\phi}}_{{computed}}^{}$|₂$\left.\vphantom{\Vert \hat{\phi}_{exact} - \hat{\phi}_{computed} \Vert _2 }\right/$|$\hat{{\phi}}_{{exact}}^{}$|₂ . The errors are plotted as a function of average cell length in Fig. 8 together with a linear fit to the logarithm of the error as a function of the logarithm of the average cell length.

Figure 8: Plot of convergence data and least-squares fit to data.

The slope of this linear function is 1.98. Perfect second-order convergence corresponds to a slope of 2.0. Thus our support operators diffusion scheme converges with second-order accuracy for the two-material problem on random meshes.

The problem associated with the second set of calculations can be described as follows:

$${\frac{{\partial \phi}}{{\partial z}}}$$ (74)

for z $\in$ [0, 1] , with Marshak vacuum boundary conditions at z = 0 and z = 1 , and where D = ${\frac{{1}}{{30}}}$ , and Q = 1 . We refer to this problem as the homogeneous problem. The homogeneous problem is solved in 3-D on a unit cube by having the vacuum boundary conditions on two opposing sides of the cube with reflecting conditions on the remaining four sides. We have performed calculations for this problem using both random and Kershaw-squared meshes in conjunction with two different solution techniques. The first is to apply row and column scaling to the coefficient matrix and then solve the resulting system using the conjugate-gradient method in conjunction with symmetric successive over-relaxation (SSOR) for preconditioning. We refer to this as the one-level solution technique. The second is to apply row and column scaling to the coefficient matrix and then solve the resulting system using the conjugate-gradient method in conjunction with the low-order 7-point cell-center diffusion scheme for preconditioning. We refer to this as the two-level solution technique. The low-order equations are solved by first applying row and column scaling to the low-order coefficient matrix and then using the conjugate-gradient method in conjunction with SSOR preconditioning. Note that the low-order system is solved once per full-system conjugate gradient iteration. The total conjugate-gradient iterations required for the full system, the maximum iterations required for the low-order system, and the total CPU time is given for each calculation in Table I.

Table I: Comparison of One-Level and Two-Level Solution Techniques.

Technique	Mesh Type	FS	Max LO	CPU Time
		Iterations	Iterations	(Sec)
One-Level	Random	97	-	143.24
Two-Level	Random	7	32	61.53
One-Level	Kershaw2	175	-	247.17
Two Level	Kershaw2	46	42	352.91

It can be seen from Table I that the two-level solution technique takes 14 times fewer full-system iterations than the one-level solution technique on the random mesh, but it takes only about 3.5 times fewer full-system iterations on the Kershaw-squared mesh. This is expected since the low-order scheme becomes increasingly inaccurate relative to the full scheme as the mesh becomes increasingly skewed. The reduction in iterations observed for the random mesh is indicative of the reduction seen in well-formed meshes. Note that the two-level scheme is faster than the one-level scheme on the random mesh, but it is slower than the one-level scheme on the Kershaw-squared mesh. The decrease in CPU time for the two-level scheme will be very dependent upon the method used to solve the low-order system. For instance, rather than solve the low-order system to a high level of precision using a Krylov method, one might simply perform a fixed number of multigrid V-cycles. This would greatly reduce the cost of the preconditioning step and thereby reduce the total CPU time as well. Such a strategy was employed with great benefit in [1]. It is important to realize that the structure of the low-order cell-center system on structured meshes is compatible with standard multigrid methods such as Dendy's method [14], whereas the full system has a structure that is incompatible with standard methods. Thus the low-order preconditioning approach enables highly efficient solution techniques to be used in an indirect manner when they cannot be directly applied to the full system.