Spectrum of the Galerkin 2D Convective Operator

We can now examine the behaviour of this operator by examining its eigenspectrum. The distribution of the spectral radius, $\rho(\theta)$, shows us the level of directional inhomogeneity of wave frequency supported in a given domain. We first consider a periodic box that is discretized with essentially standard elements, and then we examine how deforming these elements within the box affects wave propagation.

  

Figure 4.9: Spectral radius of the weak convective operator on a periodic domain, N=12.

In figure 4.9 we show three discretizations of the periodic box. The first one (a) employs only quadrilaterals, the second one (b) a mix of quadrilaterals and triangles, and the last one (c) only triangles. $G(\theta)$ was constructed using a 12^th order expansion and the spectral radius of the $G(\theta)$ for each mesh is shown as a function of $\theta$. In the upper right quadrant we see that the spectral radii are very similar for all three cases, but in the lower right quadrant we see that there is a marked difference between the spectral radii of the quadrilateral mesh and the triangle mesh with the hybrid mesh between these two cases.

Theoretically, we do not have to consider the spectral radius of the mixed basis, as we have shown that since it is numerically similar to the modal basis it will share the same spectral properties for linear operators. However, we did run this test for the quadrilaterals using a nodal basis and obtained the same spectral radius to machine precision, confirming the theory.

  

Figure 4.10: Growth of the spectral radius of the Galerkin convective operator with expansion order.

  

Figure 4.11: $\rho_\Box(\theta)$ is an exact fit for the spectral radius of the Galerkin convective operator on a periodic domain tiled with regular quadrilaterals and with N=12.

  

Figure 4.12: $\rho_\triangle(\theta)$ is a close fit for the spectral radius of the Galerkin convective operator on a periodic domain tiled with triangles and with N=12.

So far we have examined the spatial variation at a given expansion order. In figure 4.10 we now demonstrate that $\sup_{\theta} \rho(G(\theta))$ grows as O(N²) for the modal basis used on all three meshes, and again we note that the mixed basis has exactly the same property to machine precision.

  

Figure 4.13: Spectral radius of the Galerkin convective operator on a periodic domain discretized with non-regular elements.

We have an exact fit for the numerical spectral envelope of the Galerkin convective operator on a periodic square domain discretized with regular quadrilaterals. This can be represented as (see also figure 4.11):

$$\rho_\Box (\theta) = \left\{ \begin{array} {ll} \sqrt{2} \r... ...heta+\frac{3\pi}{4}), & \mbox{otherwise.} \end{array} \right.$$

We also have an approximate fit for the numerical spectral envelope of the Galerkin convective operator on a periodic square domain discretized with regular triangles. This is described by (see also figure 4.12):

$$\rho_\triangle (\theta) \approx \left\{ \begin{array} {ll} ... ...Box(0)) sin(2\theta)), & \mbox{otherwise} \end{array} \right.$$

We have not so far indicated how the operator $G(\theta)$ depends on the skewness of the physical elements. We will now consider the same periodic box discretized with deformed elements, so that these mappings are not simply scaled identities. In figure 4.13 we have created deformed elements by simply shifting the vertex in the middle of the quadrilateral and triangle meshes we have just considered. In addition, we created a new triangle mesh by taking the Delaunay triangulization of the given vertices. This time we see that the deformation of the triangle mesh has increased the spectral inhomogeneity of the operator, but that this can be ameliorated by choosing the Delaunay triangulization.