# Fast and Efficient Lunar Finite Element Gravity Model
## 3 Representations of Lunar Potential Using Orthogonal Finite Elements
### Results of the FEM LP Analysis
For the proposed lunar model, it is found that the most efficient maximum approximation error is \(10^{-9}\) m s\({}^{-2}\), which provides a fair balance between computational time and machine precision at a maximum polynomial order of \(N=18\) on the lunar surface with a \((4\times 4)\) square degree FEM cell size. However, when a polynomial order of 18 was applied to a radius 100 km above the lunar surface, the error produced was negligible. This confirmed the previous discovery about a decreasing radial polynomial order, i.e., as you go further from the surface, the polynomial order required to reach the error tolerance decreases as well. Furthermore, a maximum degree value (\(M\)) of 100 and a shell value of \((4\times 4)\) degree maintains an efficient error tolerance while constraining sensible computational power. The determination of the correct polynomial order for each radial adaptation allowed for us to determine the number of coefficients required in approximation at each radius. At the lunar surface, the polynomial order of \(N\,=\,18\) indicates that the gravity acceleration at the lunar surface can be approximated by \((18+1)^{2}=361\) orthogonal polynomial terms (or coefficients). At 200 km above the surface, a polynomial order of \(N\,=\,6\) was found to be most efficient, which means it requires 49 coefficient terms. This suggests that as the polynomial order increases, the number of coefficients required in the approximation also increases as well. This relationship can also be supported when observing how an error is related to the node number at each radius. Figure 9 displays two plots: (a) polynomial  
Figure 8: LP Maximum Error of Chebyshev FEM Gravity Approximation (m s\({}^{-2}\)) versus Polynomial Order (N), for radial distances from \(r_{min}=R_{M}\) to \(r_{max}=1.11R_{m}\), 200 km above the surface. order as a function of radius and (b) number of coefficients as a function of radius; these plots show a nearly similar result. On the same machine, the FEM model displays three orders of magnitude of computational speedup at the lunar surface compared to the (100, 100) LP SH model. This can be seen in Figure 10. The FEM approach continues to be computationally efficient compared to the original (100, 100) SH model, even going as far as to decrease computational time as it reaches higher radii. At radii \(R\) near (150 km \(+\)\(R_{m}\)) \(<\)\(R\)\(<\) (200 km \(+\)\(R_{m}\)), it is found that the FEM approaches five orders of magnitude of computational efficiency compared to the SH model. If implemented for orbit propagation, the FEM gravity model should not only provide nine to ten digits of accuracy, but it also produces a three to five order of magnitude reduction in CPU time. The Chebyshev coefficients were saved for each radial iteration. These first 49 coefficient terms are displayed for the final cosine sampling, as shown in Figure 11. With cosine sampling, higher-order non-zero coefficients have been found to be comparable to easy-to-fit smooth functions of similar nature. With this, it is observed that the norm of gravity decays rapidly, as it should. Figure 12 qualitatively displays a prompt decay of gravity anomalies in a finite element where the contours are created by the gravitational acceleration perturbation at three radii located at the surface, 100 km, and 200 km above the  
Figure 9: Plots of Polynomial Order \(N\) and Number of coefficients as a function of normalized radius \(R_{m}\). (**a**) Polynomial Order as a function of radii. (**b**) Number of coefficients as a function of radii. surface. These coefficients can then be used to approximate orbital propagation at different radii on or above the lunar surface. When considering the coefficients to be a function of radius, the two-dimensional approximations on each surface are used to promote three dimensions by representing each coefficient as an orthogonal Chebyshev function of radius.