Optimized Parallel Computing for the Kolmogorov Equation, 2 Subdomains

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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, 1-sided Optimized Robin and 2-sided Optimized Robin algorithms. We observe the numbers of iterations till convergence of the algorithms in four successive dyadic mesh refinements, 2-jx0.01 with j=0,..,4.

 

In the overlapping case, both OSWR(p) (1-sided Robin) and OSWR(p,q) (2-sided Robin) algorithms appear to be almost insensitive to the mesh refinement, while the CSWR appears to be very sensitive to it. The two-sided OSWR(p,q) appears globally more robust in terms of iteration counts with respect to the one-sided OSWR(p), whose iteration counts still remain more than reasonable. Both algorithms outperform the CSWR. In the non-overlapping 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.