Triangle Basis: ${\bf TrNH}$

We will discuss four ways that the triangle basis can be constructed, but eliminate three of them due to undesirable properties of the resulting basis. A first approach might be to construct the triangle basis in the same way as the quadrilateral, using a complete set of N² tensor products of the Legendre interpolant functions $\{h_n\}$. This basis suffers from over-resolution at the singular vertex of the coordinate system as shown in [16]. To avoid this overresolution we could choose a subset of the Legendre interpolant functions on T², as suggested in [34]. This leads however to a basis that suffers from numerical linear dependency as noted in [16]. Lastly, we could construct a two-dimensional nodal basis in the manner of [35] or [36] by explicitly specifying the node distribution, but these bases do not have the tensor product property thus making spatial derivatives expensive O(N⁴) operations. Recently, however, fast polar derivative techniques have been developed [37] that reduce the scaling factor in the asymptotic cost. These advances have made the cost comparable to the tensor product cost for N < 10. It is still true that a modal approach provides a natural way to to use exact integration and thus avoid aliasing.

Instead, we propose a basis that is compatible with the nodal quadrilaterals (${\bf QuNN}$) and numerically similar to the modal triangle basis (${\bf TrHH}$). The C⁰ continuity condition requires that the boundary modes of the triangle must have the same shape as the $\{h_n\}$ modes of the quadrilaterals they can share an edge with. However, this condition does not determine how the modes should be shaped in the interior of the triangle, which leaves us a certain amount of freedom. We require that the modes are polynomials in (r,s) and that the basis spans the polynomial ${\cal P}_{N}$. We complete the requirement that the bases be numerically similar by ensuring the new basis is numerically linearly independent. This is guaranteed by using the same interior modes as the modal triangle basis. As previously discussed, the choice of $(\alpha,\beta)$ for the Jacobi polynomials is key in ensuring that the interior modes do not degenerate in the (r,s) coordinate system but are still tensor products in the (a,b) coordinates.

We constructed the vertex and edge modes of the new basis in the (r,s) coordinates. The vertex 3, edge 2, edge 3, and interior modes still maintain tensor-product in the (a,b) coordinates but the remaining boundary modes become two-dimensional functions in this frame. These new modes will increase the constant factor in the asymptotic cost of inner products and transforms compared to the modal triangle.

The new basis is presented in closed form, but we note that it could also be derived as a linear combination of modes from the modal basis, and this is how we guarantee that it will share all properties that the modal basis has for linear operations.

So we now present the nodal-compatible ${\bf TrNH}$ basis for the triangle in local cartesian coordinates: Vertex Modes:

Edge Modes $( 2 < m,n \leq N)$:

In standard triangular coordinates the vertex Modes are:

Edge Modes (2 < m,n < N):

The interior modes are unchanged from the modal triangle basis. We represent this basis for N=5 in figure 3.8; the interior modes still have the bubble shape and thus they are zero at the boundaries.

  

Figure 3.8: Mode shapes for the triangle mixed basis (${\bf TrNH}$) with N=5.