I'm interested in calculating resonance values using PML for scattering equations. In a basic case I consider the unit circle and assume a constant valued scatterer n. My goal is to verify a resonance value that I calculated by hand before modifying the code for more interesting shapes and scatterers. The equation of interest is given by:
[tex]\Delta u + (1+n)\omega^2 u =0[/tex]
with Sommerfeld radiation condition
[tex]\lim_{r \to \infty} r^{1/2}
\bigg(
\frac{\partial u }{ \partial r} - i \omega u
\bigg) = 0[/tex]
to calculate the resonance values I can modify the code in the tutorial to add the unit circle along with defining a PML region. The problem I'm having is that with the radiation condition I have a boundary term in the weak formulation. This is seen in the line of code from 1.7.1 tutorial by
Code:
a = BilinearForm(fes)
a += grad(u)*grad(v)*dx - omega**2*u*v*dx
a += -1j*omega*u*v*ds("outerbnd")
The issue is that the example to calculate resonances does not have this boundary term. Ideally I'm trying to solve the following integral equation for [tex]\lambda = \omega^2[/tex] but I do not believe the Arnoldi solver will suffice:
[tex]\int_\Omega\big[ \nabla u \cdot \nabla \bar v \big]
\, dx -
i \,\omega\, \int_{\partial \Omega} u \bar v \, ds = \int_{\Omega} \omega^2 u \bar v[/tex]
Could anyone provide some insight as to how I would solve this problem?
