Results

The problem was solved using both the approaches mentioned above. To study the effect of the size of the grid on the convergence rate, several runs were conducted with increasing grid sizes and the number of iterations required to converge to a fixed state (where the residual was two orders lower ,i.e., 100 times smaller than the initial residual) was recorded. The data for both the approaches can be seen in figure 1. A polynomial fit was done on the data, and it was observed that a quadratic curve fits the data almost perfectly. The number of iterations required for an N x N grid for version 1 of the program can be represented as 0.890 N² + 1.97N+0.06, and the same for version 2 of the program can be represented as 0.445 N²+1.00N + 0.67. A sample convergence plot ( $\log_{10} \vert\vert\textrm{Frobenius norm}\vert\vert$ vs number of iterations) for a $50\times50$ grid for both the approaches can be seen in figure 2.

Figure: Effect of the grid size on the number of iterations required (for $\log_{10} \vert\vert\textrm{Residual}\vert\vert = -2$)

Figure: Convergence rate for a $50\times50$ grid