We have performed a set of calculations intended to demonstrate that
our support-operators method converges with second-order accuracy for a problem with a material discontinuity and a non-smooth mesh.
There are two types of meshes used in all of the calculations: orthogonal and random.
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 r_{0}
, where
r_{0} = 0.25l_{c}
. These random meshes are both non-smooth and skewed, but these properties are approximately constant independent of the mesh size.
The problem associated with the first set of calculations can be described as follows:
- D(z) = Qz^{2} ,
for
z [0, 1]
, where
D(z)
=
D_{1} ,,
=
D_{2} ,,
with an extrapolated zero intensity at
z = 1 + 2D
and z = - 2D
,
and where
D_{1} =
,
D_{2} =
, and Q = 1
.
The exact solution to this two-material problem is:
=
a + bz + c_{1}z^{4} ,,
=
a + c_{2}z^{4} ,,
where
a = , b = ,
c_{1} = - , c_{2} = - .
This problem is solved in 3-D on a unit cube having the extrapolated 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 preceeding calculation.
The relative L_{2}
intensity error was computed for each calculation and is plotted as a function of average cell length in Fig.5 together with a linear fit to the logarithm of the error as a function of the logarithm of the average cell length.
The slope of this linear function is 1.98.
Perfect second-order convergence corresponds to a slope of 2.0.
Thus we conclude that our support operators diffusion scheme converges with second-order accuracy for the two-material problem on random meshes.
Figure 5: Logarithmic Plot of Error Versus Cell Width