# Fast and Efficient Lunar Finite Element Gravity Model
## 3 Representations of Lunar Potential Using Orthogonal Finite Elements  
In the following discussion, we consider a basic orthogonal FEM approximation to a gravitational acceleration model determined from NASA's Lunar Prospector (LP) mission in 100 spherical harmonic degrees [48]. Released in 2000, the LP Gravity model allows considerations of an analogous FEM approximation of the 'LP100K' gravity model. For high-precision orbit propagation, all terms used in the \((m,n)~{}=~{}(100,100)\) LP model are considered. Using this model, [PERSON]'s equation is solved for gravity using FEM and compared to the classical spherical harmonic approximations. Initially, a test was conducted to produce a contour plot of the disturbance gravity acceleration on a small degree order using the FEM approach. This test ensured a reasonable error value on top of a reasonable computation time, which encouraged us to continue the procedure with a higher degree order for an accurate representation. Figure 4 shows the radial disturbance acceleration on the lunar surface from the FEM representation with \((4\times 4)\) degree square using the \(N=10\) approximation.  
Figure 4: Radial FEM gravity approximation at the lunar Surface for LP (\(100\times 100\)) ( m s\({}^{-2}\)).  
Figure 3: Orthogonal approximation routine in n-th dimensions.  
As mentioned in the introduction, the three main challenges faced when using SH for orthogonal approximations are (1) an upper limit discrepancy, (2) inefficiency at orders higher than 2, and (3) non-free singularities at the poles of the astronomical bodies. With these three challenges in mind, a finite element method model of surface gravity representations is considered in the hopes that lower degree functions produced can be implemented to effectively and sufficiently model and compute lunar gravity. Furthermore, this method can be parallelized to increase computational efficiency. With this, the FEM model is structured to be both radially adaptive and efficient, which will result in better interpolation at higher degrees (i.e., at different radial distances above the surface of the Moon).  
The following gravitational potential model is made up of the reference and disturbance gravity terms where the reference term includes two-body effects on the lunar gravity (\(U_{2B}\)), while the disturbance term (\(\Delta U(r,\lambda,\phi)\)) includes any other effects on gravity for the higher degree and order terms to take place. With these two terms, the finalized potential, \(U(r,\lambda,\phi)\), can be given as:  
\[U(r,\lambda,\phi)=\underbrace{U_{2B}(r,\lambda,\phi)}_{Reference\ Term}+ \underbrace{\Delta U(r,\lambda,\phi)}_{Disturbance\ Term} \tag{26}\]  
Furthermore, the acceleration can be derived from the potential and is represented as:  
\[\ddot{r}=-\frac{\partial U}{\partial r}=\underbrace{\frac{\partial}{\partial r }\left\{\frac{GM}{r}\right\}}_{Reference\ Terms}-\underbrace{\frac{\partial}{ \partial r}\Delta U}_{Disturbance\ Term} \tag{27}\]  
It is important to note that adjusting the model to correct for the two-body interference is performed before the reference potential compact and efficiency are considered. This means only the perturbing acceleration terms are considered in the model.