## 5 Measured Results
### Mesh Size Testing Results  
Table 11 shows the time taken and iterations needed before converging to a solution using the Python script based on mesh sizes of 9, 17, 33, 65, 129, 257, and 513 with the same aforementioned boundary conditions. The recorded number of iterations needed is expressed as a logarithmic function as shown in Figure 10.  
In Figure 10, a noticeable trend emerges revealing a significant rise in convergence time as the mesh size surpasses 129. Opting for a mesh size of 129 or higher promises heightened accuracy, particularly when time constraints are lenient, though at the expense of increased computational duration. For expedited solutions, a mesh size of 65 proves \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{6}{c}{**T(x,y) in \({}^{\circ}\)C**} \\ \hline \multicolumn{2}{c}{**Excel (Analytical Solution)**} & \multicolumn{3}{c}{**Python (Numerical Solution)**} & \multicolumn{3}{c}{**Difference/(\%) (Excel—Python Values)**} \\ \hline \multicolumn{2}{c}{**(0,0)**} & \multicolumn{1}{c}{**(0.25,0)**} & \multicolumn{1}{c}{**(0,0)**} & \multicolumn{1}{c}{**(0.25,0)**} & \multicolumn{1}{c}{**(0.5,0)**} & \multicolumn{1}{c}{**(0,0)**} & \multicolumn{1}{c}{**(0.25,0)**} & \multicolumn{1}{c}{**(0.5,0)**} \\ \hline \multicolumn{2}{c}{**(0,0.25)**} & \multicolumn{1}{c}{486.4} & 480.9 & \multicolumn{1}{c}{**(0,0.25)**} & 485.7 & 480.4 & \multicolumn{1}{c}{**(0,0.25)**} & \multicolumn{1}{c}{0.7/(0.14\%)} & \multicolumn{1}{c}{0.5/(0.10\%)} \\ \hline \multicolumn{2}{c}{**(0,0.5)**} & \multicolumn{1}{c}{463.6} & 450 & \multicolumn{1}{c}{**(0,0.5)**} & 462.5 & 450 & \multicolumn{1}{c}{**(0,0.5)**} & \multicolumn{1}{c}{1.1/(0.24\%)} & \multicolumn{1}{c}{0/(0.0\%)} \\ \hline \multicolumn{2}{c}{**(0,0.75)**} & \multicolumn{1}{c}{413.6} & 391.9 & \multicolumn{1}{c}{**(0,0.75)**} & 414.3 & 394.6 & \multicolumn{1}{c}{**(0,0.75)**} & \multicolumn{1}{c}{\(-0.7/(-0.17\%)\)} & \multicolumn{1}{c}{\(-2.7/(-0.69\%)\)} \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Difference between analytical and numerical solutions. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Mesh Size**} & \multicolumn{1}{c}{**Iterations Needed**} & \multicolumn{1}{c}{**Time Taken (s)**} \\ \hline \multicolumn{2}{c}{**9**} & 169 & 0.0327 \\ \hline \multicolumn{2}{c}{**17**} & 613 & 0.262 \\ \hline \multicolumn{2}{c}{**33**} & 2171 & 3.63 \\ \hline \multicolumn{2}{c}{**65**} & 7537 & 50.0 \\ \hline \multicolumn{2}{c}{**129**} & 25,547 & 684 \\ \hline \multicolumn{2}{c}{**257**} & 83,781 & 8998 \\ \hline \multicolumn{2}{c}{**513**} & 261,485 & 112,748 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Iterations needed against time taken for varying mesh sizes. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{_T(x,y)_ in \({}^{\circ}\)C**} \\ \hline \multicolumn{2}{c}{**(0,0)**} & \multicolumn{1}{c}{**(0.25,0)**} & \multicolumn{1}{c}{**(0.5,0)**} \\ \hline \multicolumn{2}{c}{**(0,0.25)**} & \multicolumn{1}{c}{485.7} & 480.4 \\ \hline \multicolumn{2}{c}{**(0,0.5)**} & 462.5 & 450 \\ \hline \multicolumn{2}{c}{**(0,0.75)**} & 414.3 & 394.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Numerical solutions obtained via Python. optimal, consuming a mere 50 s compared to the considerably longer 684 s required for larger mesh sizes. Figure 11 corroborates these findings by showcasing distinct disparities in contour plots generated across varying mesh sizes, highlighting the enhanced precision achieved as temperature gradient lines exhibit smoother convergence towards the intended corners. Notably, the discrepancy between contour plots generated by mesh sizes of 65 and 513 is marginal, rendering the additional computational overhead unjustifiable. Consequently, a mesh size of 65 is determined as the most efficient choice going forward, striking a balance between accuracy and computational efficiency. For better clarity for Figure 11, you can refer to the following:  
Mesh Size 9, Header: 20 Steady-State Conduction Equation (Gauss-Seidel Method) Iterations Number = 169.