I'm looking to solve Laplace's equation on a polygon with Dirichlet and homogenous Neumann conditions using Schwarz-Christoffel (CS) mapping.
I'm able to map the polygon to the upper-half plane using using the standard CS map:
$$ f(z) = A\int_{0}^{z} \prod_{i=1}^{n-1}(\zeta - x_i)^{\alpha_1}d\zeta + B $$
Now the Dirichlet and Neumann boundary conditions correspond to intervals on the real line seperated by the pre-vertices $x_i$.
A second SC map can be used to take the upper-half plane to a region where the Dirichlet intervals are mapped to vertical lines and the Neumann intervals are mapped to horizontal lines (Driscoll and Trefethen Schwarz-Christoffel Mapping, 2002). The harmonic function on the half plane is the real part of this second SC map. For the case of alternating Dirichlet and Neumann intervals, the desired harmonic function can be found with the following theorem:
Let $\kappa >1$, real values $\phi_1, ..., \phi_{\kappa}$ and disjoint real intervals $$ I_1 = (-\infty, x_1)\\ I_2 = (u_2, x_2)\\ \vdots\\ I_{\kappa} = (u_{\kappa}, x_{\kappa}) x_{\kappa} < \infty $$ with $x_{\kappa} < \infty$ be given. Then there is a unique real polynomial $p(z)$ of degree no greater than $\kappa - 2$ such that $$ \phi(z) = \phi_1 + \text{Re} \Bigg[ \int_{x_1}^{z} p(\zeta)(\zeta - x_1)^{-1/2}\prod_{j=2}^{\kappa}(\zeta - u_j)^{-1/2}(\zeta - x_j)^{-1/2} \Bigg] $$ is the unique harmonic function in the upper half plane such that $\phi(x) = \phi_j$ if $x \in I_j$ and $\partial \phi/ \partial n = 0$ for $x \notin I_j$.
If we define $$ g(\zeta) = (\zeta - x_1)^{-1/2}\prod_{j=2}^{\kappa}(\zeta - u_j)^{-1/2}(\zeta - x_j)^{-1/2} $$ then the polynomial $p(\zeta)$ is given by $$ p(\zeta) = b_0 + b_1\zeta + .. + b_{\kappa-2}\zeta^{\kappa-2} $$
where $b_0, b_1, ..., b_{\kappa-2}$ are the solution to the linear problem $$ M \begin{bmatrix} b_0\\ b_1\\ \vdots\\ b_{\kappa-2} \end{bmatrix} = \begin{bmatrix} \phi_2 - \phi_1\\ \phi_3 - \phi2\\ \vdots\\ \phi_{\kappa} - \phi_{\kappa-1} \end{bmatrix} $$ and M is given by $$ \begin{bmatrix} \int_{x_1}^{u_2} g(\zeta) d\zeta & \int_{x_1}^{u_2}\zeta g(\zeta) d\zeta & ... & \int_{x_1}^{u_2}\zeta^{\kappa - 2} g(\zeta) d\zeta\\ \int_{x_2}^{u_3} g(\zeta) d\zeta & \int_{x_2}^{u_3}\zeta g(\zeta) d\zeta & ... & \int_{x_2}^{u_3}\zeta^{\kappa - 2} g(\zeta) d\zeta\\ \vdots & \vdots & \ddots & \vdots\\ \int_{x_{\kappa-1}}^{u_{\kappa}} g(\zeta) d\zeta & \int_{\kappa-1}^{u_{\kappa}}\zeta g(\zeta) d\zeta & ... & \int_{\kappa-1}^{u_{\kappa}}\zeta^{\kappa - 2} g(\zeta) d\zeta\\ \end{bmatrix} $$
Now, my question is if there is any way to use this for the case of non-alternating Dirichlet and Neumann conditions. For my situation, I have only one Neumann condition with the rest Dirichlet conditions. By setting the Neumann intervals in the above formulation to 0, the matrix M becomes singular so the coefficients of $p$ cannot be found. A remark in Driscoll and Trefethen's Schwarz-Christoffel Mapping says that the solution to the piecewise-constant boundary value problem can be obtained by letting $x_{j} \to u_{\kappa+1}$ for $j < \kappa$ and $x_{\kappa} \to \infty$. Can we do this for only some of the intervals for a general non-alternating Dirichlet and Neumann boundary value problem?
If it's not possible, suggestions for an alternative way to map the upper half plane to a convenient domain where $\phi$ could be found would be welcome. Or any other way to analytically find $\phi$ in the upper half plane with mostly Dirichlet and one homogenous Neumann boundary condition.