6.5. Exercises for perfectly matched layers

Exercise 1

Write down the necessary single steps for deriving and implementing a Perfectly Matched Layer for the Helmholtz equation. Indicate which of these steps are necessary for the implementation and which are purely theoretical. Which of the steps do you expect to introduce an error?

Exercise 2

Use the addition of PML objects to generate a two-dimensional cartesian scaling with different damping in \(x\) and \(y\) direction. Use pml.Compound to combine suitable scalings to a cylindrical complex scaling, i.e., radial in the \(x,y\) plane and cartesian in the \(z\) direction. What kind of mesh is necessary to resolve the kinks generated by this PML?

Exercise 3

Implement a PML for the two dimensional waveguide problem with an initial Gaussian peak by manually setting the complex coefficients. As an alternative implement it using pml.HalfSpace.

Exercise 4

Solve the Helmholtz resonance problem on the geometry square_inf.ipynb surrounded by a PML (radial or cartesian). Try to explain the resulting discrete spectrum.