Helsinki University of Technology Somersalo/Bingham Mathematics

Mat-1.198 Scattering Theory

9^{th} set of exercises, 9.4.2003

The exam will be on Tuesday, the 20th of May from 10.00 to 13.00 in U322.

1. Consider the one-dimensional potential scattering of the previous week. Derive the PML equation in the one-dimensional case: The stretching function

F : ℝ → ℂ, x ↦ \{ \begin{matrix}x − iτ(a − x), & x < a \\ x, & a ≤ |x| ≤ b \\ x + iτ(x − b), & x > b \end{matrix}

maps the real axis to a curve in the complex plane. Here (a,b) ⊃ [−M,M] is the truncated computation domain.

2. Consider the PML equation in radial coordinates, i.e., the computational region B is a

disc,

\overline{D}⊂ B = \{x ∈ ℝ^{2} | |x| < R\} ⊂ ℝ^{2}.

Given a stretching function τ: [0,∞) → [0,∞), calculate the PML equation using the polar coordinate representation for the Laplacian.

3. The previous example allows us to write the PML equation as

∇ · A∇u + ak^{2}u = 0,

where A = A(x) ∈ ℂ^{2×2} , a = a(x) ∈ ℂ. Find this representation. In particular, what is A in Cartesian coordinates?