Second-Order Demonstration

Support Operator Method:

Problem Size (cells)	${\frac{{\left\Vert \Phi_{\mbox{\scriptsize exact}} - \Phi \right\Vert _2}}{{\left\Vert \Phi_{\mbox{\scriptsize exact}} \right\Vert _2}}}$	Error Ratio
2 x 2 x 2	7.4950 x 10-2
4 x 4 x 4	2.4163 x 10-2	3.10
8 x 8 x 8	5.5245 x 10-3	4.37
16 x 16 x 16	1.5467 x 10-3	3.57
32 x 32 x 32	3.6797 x 10-4	4.20
64 x 64 x 64	9.6113 x 10-5	3.82

Morel-Hall Asymmetric Method:

Problem Size (cells)	${\frac{{\left\Vert \Phi_{\mbox{\scriptsize exact}} - \Phi \right\Vert _2}}{{\left\Vert \Phi_{\mbox{\scriptsize exact}} \right\Vert _2}}}$	Error Ratio
2 x 2 x 2	7.4350 x 10-2
4 x 4 x 4	2.4044 x 10-2	3.09
8 x 8 x 8	5.4575 x 10-3	4.41
16 x 16 x 16	1.5256 x 10-3	3.58
32 x 32 x 32	3.6960 x 10-4	4.12
64 x 64 x 64	9.5032 x 10-5	3.88

Two-material problem, ratio = 10, GMRES/CG, Low-Order Preconditioner, $\epsilon$ = 10^-10 , $\epsilon_{{pre}}^{}$ = 10^-9