Connected Dirichlet and Neumann conditions Im working on a project where we have to solve (numerically) the temperature distribution in an apartment, using Laplace equation. The following is a simplified model for that apartment, which illustrates the question i have. The numbers are the constant temperaturs of the walls. 
So, there are 2 rooms, connected with an interface. Here comes the part which confuses me:

Prescribe dirichlet condition on the interface of the left room, and neumann condition on the interface of the right room. 

Im familiar with those conditions, but not in this context. There is no info given about the values of the temperature neither the derivative, at those interfaces, so how should i think about this? So, i have 2 questions:


*

*What phisical interpretation would this model explain? What kind of "wall" would that interface be?

*I want to solve this numerically, so i would start with a guess, and then iterate until convergence. What conditions should i prescribe at the interface for each room?
 A: Essentially, this is simply applying "continuity" to the two rooms. If you assume that the wall can't store heat (a slight simplification) or that it's not a real wall, then any flow of heat from the left room through the wall must enter the right room, and vice versa.
Similarly, thermal contact means that the temperature of the wall (or the points on the boundary between the two rooms) must be the same (neglecting wall thickness) on both sides of the wall.
However, you can't apply both requirements on one side of the wall. So instead, you apply one condition on each, and turn the problem into a "matching" problem - that is, the solutions need to "match" at the wall.
The logic works like this:
Solve in the left room assuming that the temperature on the wall is $T_w(y)$ (or $T_w(y,z)$, if working in 3D). This will give you a solution in the left room dependent on that temperature on the wall. Let's call this $T_l(x,y;T_w)$.
Now, solve in the right room assuming that the gradient of the temperature at the wall in the right room is $G(y)$. You will get another solution, of the form $T_r(x,y;G)$.
Now, you solve for $T_w$ and $G$ by requiring that $T_r(0,y;G)=T_w$ and $\frac{\partial T_l}{\partial x}(0,y;T_w) = G$, assuming the wall is at $x=0$. By "matching" the two solutions, you get the full solution.
Solving for Dirichlet on one side and Neumann for the other is simply a matter of logic; it doesn't come from the physics, but from convenience.

When working numerically, you can solve it by iterating between the sides. However, you may find it easier to simply ignore the wall and solve on the entire domain, as the geometry of the room and boundaries is dealt with by the mesh, and doesn't need separate handling for this boundary, unless the wall has more complicated behaviour (such as partial thermal storage resulting in the gradients not necessarily being the same).
