Finite Element Method with Transparent Boundary Condition
To handle unbounded computational domains, we introduce Dirichlet-to-Neumann map (DtN) which maps the Dirichlet value to the Neumann value of the solution of a boundary value problem. In our problem, we construct the DtN map based on the Rayleigh expansion radiation conditon. Then the DtN map give rise to a transparent boundary conditon on the artificial boundary which truncates the unbounded domain.
DtN map and transparent boundary condition
The DtN map is defined on the exterior domain $G_{b} = \{(x_{1}, x_{2}) \in \mathbb{R}^{2} : x_{1} \in (0, p) \text{ and } x_{2} \in (b, +\infty) \}$ with boundary $\Gamma_{b} = \{(x_{1}, x_{2}) \in \mathbb{R}^{2} : x_{2} = b\}$. The mapping $T$ takes Dirichlet value $f \in C^{\infty}$ to the corresponding Neumann value $Tf = \frac{\partial v}{\partial x_{2}}\vert_{\Gamma_{b}}$, where $v$ is the solution to the Helmholtz equation in $G_{b}$ satisfying the Dirichlet boundary condition $v = f$ on $\Gamma_{b}$ and the Rayleigh expansion radiation condition. We conclude the following boundary value problem
\[\begin{equation*} \begin{cases} \Delta v + k^{2}v = 0, \\ v = f, \\ \end{cases} \end{equation*}\]