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Is there an analog of perfectly matched layers for finite element methods? References or small examples are much appreciated.

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A perfectly matched layer (PML) is generally introduced as a continuous concept without reference to the discretization method. A damping term is multiplied by the advection operator: $$ \frac{\partial}{\partial x} \rightarrow \frac{\partial}{\partial x}\left(\frac{1}{1 + i\sigma/\omega}\right) $$ If $\sigma$ is positive, the imaginary component in the above equation drives the standard oscillatory solution of a wave equation to be exponentially decaying. A simple way of implementing this would just modify the discretized equation everywhere in space and then assign the value of $\sigma$ to be zero everywhere except near the boundaries.

So, a PML could be implemented in any volumetric discretization like finite volumes, finite elements or finite differences.

For an introduction, I like these notes. Also, try this paper.

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Take a look at some of the papers by my colleague Joe Pasciak on the PML and its approximation: http://www.math.tamu.edu/~joe.pasciak/

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