# Fast and Efficient Lunar Finite Element Gravity Model
## 3 Representations of Lunar Potential Using Orthogonal Finite Elements
### Sampling: Radii and Shells
#### 3.1.2 Shells and Polynomial Order  
After determining a proper radial sample, determining the required polynomial order as a function of radius is the next step. The objective is to determine the order that adaptively maintains an efficient error tolerance. The required acceleration error tolerance is determined from the polynomial order \(N\) at every radius. For example, if a 4-by-4-degree  
Figure 5: Comparison between uniform radius sampling and cosine radius sampling for 10 and 21 points.  
square area on the lunar surface is considered, the convergence error is the maximum absolute error between the LP SH model's acceleration values and the approximated acceleration from the FEM method.  
Consider a shell size of \((10\times 10)\) square degrees compared to a smaller size of \((2\times 2)\) square degrees. The \((2\times 2)\) shell size would require less computational power/time to compute all the coefficients over a small area, while the \((10\times 10)\) shell size would require a longer computational time to reach a similar error tolerance because of the larger number of coefficient values calculated. The compromise of time would be better justified with the resulting error produced. Although the larger shell size may take a longer period of time to produce results, if these results display a small error tolerance (\(10^{-9}\) m s\({}^{-1}\)), it would be best to use this. Thus, when determining the correct shell size for this model, it follows to pursue a shell size that maintains an efficient error tolerance while not compromising computational power.  
With a maximum degree order of 100 and shell sizes of \(((4\times 4)\) and \((6\times 6)\), both sizes are elevated and checked for accuracy and computational time across a portion of the near side of the lunar surface (from \(0\leq\lambda\leq 180^{\circ}\) and \(0^{\circ}\leq\phi\leq 88^{\circ}\)). As discussed previously, the smaller shell size of \((4\times 4)\) would be more desirable if the computational time is reasonable. However, if the accuracy for the \((6\times 6)\) case is reasonable compared to the \((4\times 4)\) shell case, this shell size is considered. Figure 6 displays a graph of computational time and calculation error as a function of shell size. For a shell size of \((4\times 4)\), with maximum degree order of 100, it took the machine (see machine configurations in Section 4) approximately 3627.43 s to produce the relevant FEM approximation coefficients with an error of \(6.16\times 10^{-10}\) m s\({}^{-1}\). For the \((6\times 6)\) case, it took the same machine approximately 10,653.11 s to produce the FEM approximation coefficients with an error of \(1.40\times 10^{-16}\) m s\({}^{-1}\). With the shell size of \((6\times 6)\) providing such a small error compared to typical errors found in other FEM surface gravity models produced in [21], this error is insignificantly smaller given the increased computational time. With this, it is concluded that the \((4\times 4)\) case provides the best balance between computational time and reasonable accuracy.