Dear All
I would like to solve an eigen value problem on a global domain
that consists of a sphere of radius $r=R_{\rm in}$ and a shell between this radius
and the outer radiius $r=R_{\rm out}$.
In the inner sphere the wave function $\Psi$ will be expanded in terms of real-valued spherical harmonics $Y_{lm}({\hat r}$, i.e.
$$\Psi^{\rm in} ({\bf r})= \sum_{lm} f_{lm}(r)Y_{lm}({\hat r})$$
Such radial functions are most convenient in the vicinity of a nucleus with
a nuclear potential $V(r)=-2\frac{Z}{r}$ In fenics this can be formulated as a vector valued one dimensional differential equation. The boundary condition at the origin is open due to the weight function $4 \pi r^2$ vanishing
For the outer shell I will use Lagrange finite elements on tets
using fenics. The boundary condition
at $r=R_{\rm out}$ will be of the zero Dirchlet type, i.e.
$$\Psi^{\rm out} ({\bf r})=0 \mbox{for} \vert {\bf r} \vert = R_{\rm out}$$
In order to have one eigenvalue problem for the combinded two domains, I need to prescribe the boundary values at $r=R_{\rm in}$ as follows:
$$ \Psi^{\rm out}(r=R_{\rm in},\hat r)= \sum_{lm} f_{lm}(R_{\rm in}) Y_{lm}(\hat r)$$
I would appreciate any assistance as to how this can be achieved.
Sofar I have instead prescribed a robin like logarithmic condition per spherical harmonic channel, which works in principle, but converges rather slowly.
regards
Moritz
