Quadrilateral Bases: and  Modal basis  Modal basis

Triangle Basis: ${\bf TrHH}$ 

In this and the following sections we will define a set of bases for the triangle and the quadrilateral. For clarity we will label them with the first two letters of the element they will be used on (i.e. Tr or Qu for triangle and quadrilateral respectively). The next letter will denote the type of mode shapes used on the boundary of the element. H will denote hierarchical and N will denote non-hierarchical. The last letter will denote the type of mode shapes used for the interior modes, again using N and H as before. For example, ${\bf TrNH}$ will denote a triangle Tr with non-hierarchical boundary modes N and hierarchical interior modes H.

We present here a basis which is a set of tensor products with respect to the tensor product coordinates for the triangle and polynomials with respect to the reference element coordinates. It maintains numerical linear independence up to high orders due to the construction of the interior modes from Jacobi polynomials with carefully chosen $(\alpha,\beta)$ coefficients to ensure that mode shapes do not become too similar. Increasing $\alpha$ shifts the roots of the Jacobi polynomials away from the coordinate singularity at b=1 as demonstrated in [21] and hence the modes are prevented from having the same shape at this vertex.

The form of the ${\bf TrHH}$ basis is:

Vertex Modes:

Edge Modes $(2\leq m; 1 \leq n, m < M; m+n < N)$:

 Interior Modes $(2\leq m; 1 \leq n, m < M; m+n < N)$:

 We represent this basis graphically for N=5 in figure 3.1; the highest mode is quartic.
 
 

  

Figure 3.1: Mode shapes for the triangle modal basis (${\bf TrHH}$) with N=5.

  

In the next chapter we will be considering the variational form of elliptic operators. At this point it is useful to note that the ${\bf TrHH}$ basis has a very useful property that the bandwidth of its stiffness matrix is N+1 if the modes are ordered in a specific way. The stiffness matrix is defined as the inner product of the the gradients of the modes i.e.

$$S_{ij} = (\nabla \phi_i, \nabla \phi_j)$$

where the $\phi_i$ represent the modes indexed in a given way. This indexing is important. If we list the vertex modes, then the edge modes and then the interior modes we see in figure 3.2 that the stiffness matrix is partially banded. The interior modes in this case have bandwidth 2N+1 as shown in [30]. However, if we list the modes lowest order first, i.e. linear modes, then quadratic modes, then cubic and so on, we see in figure 3.3 that the stiffness matrix becomes strongly banded with bandwidth N+1. This will become important later on when we need to invert matrices of this type since reducing the bandwidth reduces the storage and number of operations required.
 
 

  

Figure 3.2: Top: Geometric ordering for modes, Bottom: Stiffness matrix $(\nabla \phi_i, \nabla \phi_j)$ for the continuous basis for triangles (N=15) with the geometric ordering of the rows and columns.

  
 
    

Figure 3.3: Top: Polynomial degree ordering for modes, Bottom: Stiffness matrix $(\nabla \phi_i, \nabla \phi_j)$ for the continuous basis for triangles (N=15) with modes ordered by polynomial order.

 

10/24/1998