Three-Dimensional Galerkin Convective Operator Spectrums

We will now consider the same problem posed in three-dimensions. Each element type will be shown to have different spectral properties. The three-dimensional linear convective equation for u(x,y,z,t) is:

$$\frac{\partial u(x,y,z,t)}{\partial t} + Lu \equiv \frac{\p... ...z)\diff{u}{x} + b(x,y,z)\diff{u}{y} + c(x,y,z)\diff{u}{z} = 0.$$

We consider the variational form of this equation in $\Omega^k$, that is:

Find $u \in H^1(\Omega)$ such that for all $w \in H^1(\Omega)$

$$\left(\frac{\partial u}{\partial t} + a\frac{\partial u}{\pa... ...tial z},w\right) = 0, \hspace{12pt} \forall w \in H^1(\Omega).$$

Applying the Galerkin formulation so that the trial and test spaces are spanned by the same basis we obtain:

$$\left( \phi_n, \phi_m \right) \frac{d \hat{u}_m}{dt} = \lef... ... \frac{\partial \phi_m}{\partial z} \right) \right] \hat{u}_m$$

We now define two local operators B_k and $L_k(\theta)$:

where we have normalized the convection velocity appropriately so that its direction $\theta$ is defined by the scalars a,b.

Based on these definitions we now construct two new operators for the entire domain:

$$L(\theta) = \left[\begin{array} {cccc} L_1 & 0 & ... & 0 \... ..... & ... & ... & ... \ 0 & 0 & ... & L_K \end{array} \right]$$

and similarly:

$$B = \left[\begin{array} {cccc} B_1 & 0 & ... & 0 \ 0 & B_... ... & ... & ... & ... \ 0 & 0 & ... & B_K \end{array} \right].$$

As in the two-dimensional case (section 4.3) we assemble the boundary degrees of the freedom to make the operator act continuously across the element interfaces. This time the Z operator maps the globally numbered modes to the locally numbered vertex, edge and face modes.