<< NDSolve`FEM`
a = ImplicitRegion[x >= 0 && y >= 0 && x <= 1 && y <= 1, {x, y}];
b = ImplicitRegion[x >= 1/4 && y >= 1 && x <= 3/4 && y <= 2, {x, y}];
o = RegionUnion[a, b];
m = ToElementMesh[o];
m["Wireframe"]
isec = RegionIntersection[a, b];
bm = ToBoundaryMesh[o, "BoundaryElements" -> {isec}]