Hello I'm Owen and I'm new here. Recently I was working on a boundary element problem about the Laplace equation. The basic Laplace equation is as follows: $$ \left\{\begin{array}{ll} \Delta \varphi=0 & \\ \varphi=\bar{\varphi}, & \text{on the boundary } \mathrm{S}_{\varphi} \\ \mathrm{q}=\frac{\partial \varphi}{\partial \mathrm{n}}=\overline{\mathrm{q}}, & \text{on the boundary } \mathrm{S}_{\mathrm{q}} \end{array}\right. $$ where $S_{\varphi}$ denotes Dirichlet boundary, $S_{q}$ denotes Neunmann boundary. With the Green's function, we can derive a boundary integral equation like: $$ \varphi(i)=\int_{\mathrm{S}}[\mathrm{G}(i, j) \mathrm{q}(j)-\mathrm{F}(i, j) \varphi(j)] \mathrm{dS}(j), \quad \forall i \in \mathrm{V} $$ Combining this formula with BEM, we can calculate the value inside the region from the value on the boundary.
However, I can't solve the cases with non-closed lines inside, for example:A circular area has a non-closed constraint line inside, which causes the area enclosed by the circular outer contour and the constraint inner contour to be non-closed, which also makes the Green function impossible.
How should I correctly calculate the effect of the right red constraint line on the inner area.
Thanks