# SUPG Stabilization#

For streamline upwind/Petrov-Galerkin (SUPG) (see for example Clevenger and Heister [2019], John and Knobloch [2006]), we add to the weak form $$a(\cdot,\cdot)$$ the cell-wise defined weak form

$a_{\text{SUPG}} (T, \varphi) = \sum_{K \in \mathcal{T}_h} \delta_K \left( \rho C_p \frac{\partial T}{\partial t} - k \triangle T + \mathbf{\beta} \cdot \nabla T - F, \mathbf{\beta} \cdot \nabla \varphi \right)_K,$

where $$K \in \mathcal{T}_h$$ are the cells in the computation, $$\delta_K \geq 0$$ is a stabilization coefficient defined on each cell, $$\mathbf{\beta} = \rho C_p \mathbf{u}$$ is the effective advection velocity. The standard literature about SUPG does not contain $$\rho C_p$$, so it makes sense to include this in the velocity. The first argument in the inner product is the strong form of the residual of PDE, which is tested with the expression $$\mathbf{\beta} \cdot \nabla \varphi$$ representing the solution in streamline direction. We have to assume $$k$$ to be constant per cell, as we can not compute the spatial derivatives easily.

For the implementation, $$\frac{\partial T}{\partial t}$$ is replaced by the BDF2 approximation, and its terms from older timesteps and $$-F$$, are moved to the right-hand side of the PDE.

We use the parameter design presented in John and Knobloch [2006] for $$\delta_K$$:

$\delta_K = \frac{h}{2d\|\mathbf{\beta}\|_{\infty,K}} \left( \coth(Pe)-\frac{1}{Pe} \right)$

where the Peclet number is given by

$Pe = \frac{ h \| \mathbf{\beta} \|_{\infty,K}}{2 d k_\text{max}},$

$$d$$ is the polynomial degree of the temperature or composition element (typically 2), $$\coth(x) = (1+\exp(-2x)) / (1-\exp(-2x)),$$ and $$k_\text{max}=\| k \|_{\infty, K}$$ is the maximum conductivity in the cell $$K$$.

If $$Pe<1$$, the equation is diffusion-dominated and no stabilization is needed, so we set $$\delta_K=0$$. Care needs to be taken in the definition if $$\| \beta \|$$ or $$k$$ become zero:

1. If $$k$$ is zero, then $$Pe=\infty$$ and the right part of the product in the definition of $$\delta_K$$ is equal to one.

2. If $$\| \beta \|$$ is zero, $$Pe < 1$$, so we set $$\delta_K=0$$.

3. If both are zero, no stabilization is needed (the field remains constant).