A Perfectly Matched Layer (PML) is an absorbing boundary condition (ABC) which can be used to approximate free-field conditions for the numerical solution of wave equation problems.

The PML is normally applied to a PDE using the following transformation:

$ \frac{\partial }{{\partial x}} \to \frac{1}{{1 + i\frac{{\sigma (x)}}{\omega }}}\frac{\partial }{{\partial x}} $

In the above, $i = \sqrt { - 1}$ and $\sigma(x)$ is a function of position in the ABC. Now apparently

$ \frac{{\partial ^2 }}{{\partial x^2 }} \to \frac{1}{s}\frac{\partial }{{\partial x}}\left( {\frac{1}{s}\frac{\partial }{{\partial x}}} \right) $

$ s = 1 + i\frac{{\sigma (x)}}{\omega } $

But is it possible to apply a coordinate stretching in the following fashion: $ \frac{{\partial ^2 }}{{\partial x^2 }} \to \frac{1}{u}\frac{{\partial ^2 }}{{\partial x^2 }} $

The coordinate stretching performed in this fashion would be similar in function to the stretching performed by the transformation applied to $\partial /\partial x$.

Essentially what I would like to do is to apply the coordinate stretching directly to $\partial ^2 /\partial x^2$. I've looked in the PML literature for a very long time, and it seems that most interest is in the application of the coordinate stretching directly to $\partial /\partial x$.

Moreover, I can imagine the coordinate-stretching occurring in a similar fashion to stretching a rubber sheet. If the stretching is being done to the coordinates of $\partial /\partial x$, then what is happening to $\partial ^2 /\partial x^2$?