Spectrum of the Galerkin 3D Convective Operator

As before we examine the spectral radius of the convective operator as a function of the direction of the wave vector. This is now a vector in 3-space so the variation of the spectral radius is a two-dimensional surface.

In three-dimensions we have four element types to consider. The observation we wish to make is that the magnitude of the spectral radius of the Galerkin convective operator scales as N² and the directional variation is a function of the element length in a given direction. This ``length'' can be defined as the minimum length between a vertex and an opposing face in the given direction.

We first consider a box discretized with tetrahedra. The spectral radius surface is shown in 4.14. There is clearly a drop in support at the top rightmost octant and the bottom leftmost octant. It is evident that the elements all have their longest edges aligned in the direction between these octants.

  

Figure 4.14: Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with six tetrahedra and fourth-order expansion.

Next we consider a box discretized with pyramids. The spectral radius surface is shown in 4.15. The mesh has a six-way symmetry that is reflected in the symmetry of the surface. The compression of the distance from apex to base causes a slight increase in resolution in that direction.

  

Figure 4.15: Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with six pyramids and fourth-order expansion.

Next we consider a non-unit box discretized with prisms. The spectral radius surface is shown in 4.16. This is a simple two element mesh showing the effect of compressing the uniform direction of the prisms. The surface shows overresolution in the vertical direction compared to the horizontal directions. If we scale the vertical direction to match the horizontal lengths the we would see that the surface is quite uniform except for underresolution in the direction aligned with the longest triangular face edges.

  

Figure 4.16: Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with two prisms and fourth-order expansion.

Lastly we repeated the experiment with a periodic box tiled with eight hexahedra shown in figure 4.17. The surface of the spectral radius reflects the symmtries and the same form of over resolution in the diagonal directions as we saw in the quadrilateral mesh case.

  

Figure 4.17: Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with eight hexahedra and fourth-order expansion.