Join Two Element Meshes

Question

Let's say I have two element meshes defined as the following:

Needs["NDSolve`FEM`"]
a = ToElementMesh[ImplicitRegion[0 <= x <= 1/2 && 0 <= y <= 1, {x, y}]]
b = ToElementMesh[ImplicitRegion[1/2 <= x <= 1 && 0 <= y <= 1, {x, y}]]

Is there any way I can construct a mesh that is the union of these two sets of mesh elements? I haven't been able to find any such commands in the documentation, but I'd like to be able to construct "boundaries" on the inside of my mesh region (in this case, a vertical line at $x = 1/2$).

Answer 1

RegionPlot can find boundaries between implicit regions even with a small number of PlotPoints. For example, you have 4 implicit regions

ineqs = {-2 <= x <= 0 && -2 <= y <= 2 && x^2 + y^2 >= 1, 
   x <= 0 && x^2 + y^2 <= 1, x >= 0 && x^2 + y^2 <= 1, 
   0 <= x <= 2 && -2 <= y <= 2 && x^2 + y^2 >= 1};

r = RegionPlot[ineqs, {x, -2, 2}, {y, -2, 2}, PlotPoints -> 10, MaxRecursion -> 2]

Then you can construct boundary mesh. Sometimes RegionPlot produces small numerical errors so I use Round[#, 0.001] to "glue" identical points. By default, ToBoundaryMesh have option "DeleteDuplicateCoordinates" -> True so vertices with the same coordinates will be reduced to one vertex.

bmesh = ToBoundaryMesh["Coordinates" -> Round[#, 0.001] &@
    First@Cases[r, GraphicsComplex[v_, ___] :> v, ∞], 
   "BoundaryElements" -> Cases[r, Line[p_, ___] :> LineElement@Partition[p, 2, 1], ∞]];
bmesh@"Wireframe"

mesh = ToElementMesh[bmesh];
mesh["Wireframe"]

Answer 2

Would this work for you?

As I understood, your goal was a boundary in the middle of the region.

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}}]}]

bmesh["Wireframe"]

mesh = ToElementMesh[bmesh];
mesh["Wireframe"]