Solving PDE with Condition on Inside of Region

Question

I'm trying to figure out how to use NDSolve to numerically solve PDEs where some internal values may be specified on the region of the solution. An example, which I picked from a textbook, is solving Laplace's equation $\nabla^2 f = 0$ on a region where we have Dirichlet boundary conditions on each edge, but we also have this set of points inside the region where $f = 0$.

In this particular case, what I would try would be to solve over each of the two halves of the box (left and right), and ensure that $f$ and $\nabla f$ are continuous over the boundary between the halves, but as you can imagine, not every problem is so symmetrical. If I use DirichletCondition to try to specify values on the interior of the region, NDSolve ignores the interior condition. Does anybody know a better way to do this?

Edit: My first guess as to how I should code this is as follows:

f[x_, y_] = NDSolveValue[{Laplacian[u[x, y], {x, y}] == 0, u[x, 0] == 0, 
 u[x, 1] == 100, u[0, y] == 100 y, u[1, y] == 100 y, 
 DirichletCondition[u[x, y] == 0, x == 1/2 && y <= 2/5]}, 
 u, {x, 0, 1}, {y, 0, 1}][x, y]

But the contour plot of this solution looks like this:

So NDSolve is clearly ignoring the internal condition.

Answer 1

I actually figured out how to do this myself from reading the answer to this question and then asking here how to construct the mesh I was looking for. I ran the following:

Needs["NDSolve`FEM`"]
bmesh = ToBoundaryMesh[
 "Coordinates" -> {{0., 0.}, {1., 0.}, {1., 1.}, {0., 1.}, {.5, 0},
  {.5,1}}, "BoundaryElements" -> {LineElement[{{1, 2}, {2, 3}, {3, 4},
  {4, 1}, {5, 6}}]}]
mesh = ToElementMesh[bmesh, "MaxCellMeasure" -> 0.0005]
f[x_, y_] = NDSolveValue[{Laplacian[u[x, y], {x, y}] == 0, u[x, 1] == 100, 
 u[x, 0] == 0, u[1, y] == 100 y, u[0, y] == 100 y, 
 DirichletCondition[u[x, y] == 0, x == 1/2 && y <= 2/5]}, 
 u, {x, y} \[Element] mesh][x, y]

...which gave me this solution:

Credit goes to the original answerers of those questions for helping me figure this out. Thanks, guys!