maelstrom.stabilization
¶
Stabilization techniques for PDEs with dominating convection. The classical article about SUPG is [BH82]; for an overview of methods, see [JK07], [JK08], and [BGS04].
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maelstrom.stabilization.
supg
(mesh, convection, diffusion, element_degree)¶ For each cell, this function return the expression
..math:
\begin{align*} \tau &= \frac{h}{2\|b\|} \left(\frac{1}{\tanh Pe} - \frac{1}{Pe}\right)\\ & = \frac{h^2}{4\varepsilon} \frac{1}{Pe} \left(\frac{1}{\tanh Pe} - \frac{1}{Pe}\right) \end{align*}
with the element diameter in the direction of the convection vector \(b\) and the Péclet number \(Pe = \frac{\|b\| h}{2\varepsilon}\); see (3) in [JK08]. Note that \(\tau\) does not have a singularity for \(\|b\|=0\) since
..math:
\frac{1}{\tanh Pe} - \frac{1}{Pe} = \frac{1}{3}Pe + O(Pe^3)
for \(Pe\approx 0\). This Taylor expansion (with a few more terms) is made use of in the code.