Optimized Parallel Computing for the
Kolmogorov Equation, 2 Subdomains

We let T=2. We decompose the [0,1]x[1,1] into two subdomains.
We discretize the subdomains by a uniform grid. We use dv=dx=dt=0.01. As a
consequence, the interface problems features 20,200 unknowns. We choose an
overlap of three elements 3dv. We initialize the interface variable with a
random value. Finally, we consider the algorithm to have converged when the
error drops below 10^{6}.
We compare the performance of the Classical, 1sided Optimized
Robin and 2sided Optimized Robin algorithms. We observe the numbers of
iterations till convergence of the algorithms in four successive dyadic mesh
refinements, 2^{j}x0.01 with j=0,..,4.
In the overlapping case, both OSWR(p) (1sided Robin) and
OSWR(p,q) (2sided Robin) algorithms appear to be almost insensitive to the
mesh refinement, while the CSWR appears to be very sensitive to it. The
twosided OSWR(p,q) appears globally more robust in terms of iteration counts
with respect to the onesided OSWR(p), whose iteration counts still remain more
than reasonable. Both algorithms outperform the CSWR. In the nonoverlapping
case, a similar pattern is observed for OSWR(p) and OSWR(p,q). Both algorithms
appear to be a little sensitive to the size of the interface problem. However,
iteration counts are higher than in the overlapping case, but not significantly
higher. The OSWR(p,q) is more robust than the OSWR(p), featuring an increase of
around 50% in iterations for the most refined case, while the latter
experiences a doubling. For both algorithms, however, the iteration counts
remain reasonable in all cases. As expected, CSWR does not converge in the
absence of overlap.