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, 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 problem associated with the first set of calculations can be described as follows:
  - D(z)$${\frac{{\partial \phi}}{{\partial z}}}$$ = Qz²   ,
  for z $\in$ [0, 1] , where
  

  $$\mbox{for $z \in [0,0.5]$}$$ D(z) =

  $$\mbox{for $z \in [0.5,1]$}$$ =

  
  with an extrapolated zero intensity at z = 1 + 2D and z = - 2D , and where D₁ = ${\frac{{1}}{{30}}}$ , D₂ = ${\frac{{1}}{{30}}}$ , and Q = 1 .

• The exact solution to this two-material problem is:
  

  $$\phi \mbox{for $z \in [0,0.5]$}$$ =

  $$\mbox{for $z \in [0.5,1.0]$}$$ =

  
  where
  a = $${\frac{{Q(1+8D_2)}}{{12D_2}}}$$   , b = $${\frac{{Q\left( D_2 - D_1 \right)}}{{192 D_1 D_2}}}$$   ,
  c₁ = - $${\frac{{Q}}{{12D_1}}}$$   , c₂ = - $${\frac{{Q}}{{12D_2}}}$$   .

• 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 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