# Solving Laplace equation with constraint on boundary

I have found the following PDE problem in a paper:

Essentially, we have a rectangular domain where there is a unknown interface $$z=\xi(x)$$ (liquid-air interface) separating the domain into two medium (air on top for $$i=a$$ and liquid at the bottom for $$i=b$$). We need to solve Laplace equation (3) subject to Dirichlet B.C. at $$z=0$$ and $$z=a+1$$ and Neumann B.C. at $$x=0$$ and $$x=\pi/k$$ (Eq.4-7). At the interface $$z=\xi(x)$$ we have the usual matching conditions for electric field and potentials. This is just a standard problem of Laplace equation if $$z=\xi(x)$$ is known. However, $$z=\xi(x)$$ is unknown, and it has to satisfy an addition equation (10). $$\tau_e$$ is the electric stress which depends on $$\phi_i$$. $$K$$ is a scalar unknown which needs to be determined as well. $$z=\xi(x)$$ is subject to Neumann B.C. (eq. 13) and an integral constraint (eq.12). So in total we need to solve for $$\phi_i(x,z)$$, $$\xi(x)$$ and $$K$$. The paper just said it is solved by FEM, but I have no idea how to set it up. Are there any general strategy for solving these kind of Laplace equation where the interface is also an unknown?

Update: added a diagram for better illustration of the problem

• I think that a diagram would help to understand your question. – nicoguaro Mar 10 at 17:26
• The general strategy is using an iterative process: assume an initial interface, solve Laplace, use that to update the interface etc. – Maxim Umansky Mar 10 at 21:44