Optimized Paralleing Computing for 2
Dimensional AdvectionDiffusion Equation, 4 Subdomains

We consider the following advection diffusion
equation on the domain [0,1]x[0,1]
The initial and boundary data are 0.
The code uses the finite element method to
solve the problem and a triangular mesh is used. The solver is GMRES. The discretization
steps in space and time are dx=dy=dt=0.01. We look only at the first iteration
in time such that T=dt. In our example, there are four subdomains (M=4) and the
decomposition in subdomains follows the x  direction. The overlapping length
is 2dx. It means that the first subdomain is [0,0.26]x[0,1], the second one is
[0.24,0.51]x[0,1], the third one is [0.49,0.76]x[0,1], and the fourth one is
[0.74,1]x[0,1].
We consider the performance of the algorithm
for several values of Robin parameter p including
small and large ones: 1, 2, 10, 20, 55. On the same figure, we also plot the
performance of the algorithm with Dirichlet transmission condition. According
to this test, the algorithm with Robin transmission conditions reach the errors
of 10^{6 }after at most 9 iterations while the one with Dirichlet
transmission conditions needs 15 iterations to reach this error.