The Decay of Basis Coefficients

Thus, each basis is a set of eigenfunctions of a singular Sturm-Liouville operator which leads us to the following observations:

Hence, if the function is infinitely smooth we see that the coefficients $\hat{u}_{ijk}$ must decrease faster than any polynomial power of i,j,k. Thus, the sum:

$$\tilde{u}^N = \sum^N_i \sum^N_j \sum^N_k \hat{u}_{ijk} \phi_{ijk}$$

must converge exponentially fast to u as N increases for all infinitely smooth u.

It is important to notice that since straight-sided tetrahedra and triangles have constant geometric mapping Jacobians these results hold for arbitrarily stretched tetrahedra and triangles. This does not follow for the other elements since their geometric Jacobians are quadratic for non-perpendicular elements. This backs up the findings that the simplical elements handle deformation better than the other types.

Unfortunately, it does not appear that this method generalizes to the pyramid in a straightforward way. However, a suitable orthogonal basis is known for a pyramid:

This basis is only appropriate for supporting Pⁿ since the `c' component is of the order i+j+k. Thus, if each $i,j,k \leq N$ but i+j+k>N then it is necessary to use high-order quadrature to integrate these modes exactly. Also, if i+j+k<N the function is a polynomial in r,s and t.