Next: Future Work Up: Morel99b Previous: Solution of the Equations

# Computational Results

• 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, lc .

• 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 = 0.25lc . 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) = Qz2   ,

for z [0, 1] , where
 D(z) = D1   ,, = D2   ,,

with an extrapolated zero intensity at z = 1 + 2D and z = - 2D , and where D1 = , D2 = , and Q = 1 .

• The exact solution to this two-material problem is:
 = a + bz + c1z4   ,, = a + c2z4   ,,

where

a =   , b =   ,

c1 = -   , c2 = -   .

• 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 L2 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

Next: Future Work Up: Morel99b Previous: Solution of the Equations
Michael L. Hall