Comparison of the Bases

  We have presented two classes of bases in sections 3.4.1 and 3.4.2 that give the same approximations and share the same spectral properties. We are left with the decision of which basis should we use for a given problem. In this section we summarize their properties and suggest possible selection criteria.

First, we consider the structure of the mass and stiffness matrices encountered in the convection and diffusion equations, respectively. We concentrate on the triangular elements only, and we examine the different structures corresponding to bases ${\bf TrHH}$ and ${\bf TrNH}$.The elemental mass matrix is the matrix which has entries defined by:

$$B_{mn} = \left( \phi_m, \phi_n \right).$$

Figure 3.9 shows in (a) the mass matrix for the ${\bf TrHH}$ basis, and in (b) the mass matrix for ${\bf TrNH}$ and in (c) the mass matrix corresponding to ${\bf TrNH}$ but calculated using exact integration. In the latter case the number of quadrature points needed is larger than the nodal points unlike the case in (b) where the quadrature points coincide with the nodes at the edges. We notice that the sparsity of the mass matrices corresponding to bases represented in (b) and (c) is reduced compared to the ${\bf TrHH}$ modal basis, but this is to be expected because each boundary mode in ${\bf TrNH}$ is a linear combination of all of the ${\bf TrHH}$ boundary modes.

  

Figure 3.9: Comparison of the elemental mass matrix for (a) ${\bf TrHH}$, (b) ${\bf TrNH}$ using collocation, and (c) ${\bf TrNH}$ using exact integration for N=15.

We can also compare the elemental stiffness matrix for the ${\bf TrHH}$modal basis and the ${\bf TrNH}$ mixed basis. The stiffness matrix is the matrix which has entries defined by:

$$S_{mn} = \left( \nabla \phi_m, \nabla \phi_n \right).$$

Figure 3.10 shows the two stiffness matrices for N=15. Again the matrix corresponding to ${\bf TrNH}$ basis has a denser structure.

  

Figure 3.10: Comparison of the elemental stiffness matrix for (a) ${\bf TrHH}$, and (b) ${\bf TrNH}$ bases for N=15.

We now summarize the properties of the two classes of bases. The modal basis (${\bf TrHH}$ and ${\bf QuHH}$) properties are:

• The basis is hierarchical.
• Gaussian integration order is not dictated by the basis order.
• We are able to vary locally the number of modes per edge or interior.
• Elemental transforms and inner products are O(N³) operations.
• The interior-interior mass matrix is banded.
• The interior-interior stiffness matrix is banded.

In addition, this basis can be enhanced with other properties by varying the form of Jacobi polynomials. For example, an interesting version proposed in [16], [32] uses $(\alpha,\beta) = (2,2)$ and (2m+3,3) for the Jacobi constants. It has the following properties:

• The interior-interior mass matrix is diagonal.
• The interior-interior stiffness matrix is full.

There are many different choices in choosing a mixed basis. Their main properties are:

• The basis is non-hierarchical (${\bf QuHN}$,${\bf TrNH}$, and ${\bf QuNN}$).
• Gaussian integration order is dictated by the basis order (${\bf QuNN}$, ${\bf TrNH}$ at the edges, and ${\bf QuHN}$ in the interior).
• The expansion order is (practically) fixed in nodal quadrilaterals (${\bf QuNN}$).
• Quadrilateral transforms and inner products cost O(N²) operations for ${\bf QuNN}$ and O(4 N²) for ${\bf QuHN}$.
• Triangle transforms and inner products are O(2N³) operations (${\bf TrNH}$).
• The interior-interior mass matrix is banded (${\bf TrNH}$); it is extremely sparse for ${\bf QuHN}$ and diagonal for ${\bf QuNN}$.
• The interior-interior stiffness matrix is banded only for ${\bf TrNH}$, otherwise it is full.

These properties suggest situations where each type of basis is appropriate: For example, the modal basis can lead to high computational efficiencies, if:

• The solution has local regions of interesting behaviour.
• The solution benefits from non-steady regions of interesting behaviour - this can be captured by local p-refinement.
• The domain is highly irregular and needs a high ratio of triangles to quadrilaterals.

On the other hand, the mixed basis can also lead to high efficiencies, if:

• The solution has a uniform variability.
• The domain is locally irregular, and thus it does not require a large number of triangles to complement the quadrilaterals.

More importantly, it is the specific application that we consider and the dynamic refinement procedure that ultimately decides which basis functions are the best choice. If it becomes apparent that an initial choice of basis becomes inappropriate, it is not an expensive operation to transform between bases.