bugs - Same region, different meshes

I'm playing around with ToElementMesh and I noticed the following behaviour that puzzles me. Consider the two regions

<< NDSolve`FEM`

Ω1 = 
  Polygon[{{-1/2, -1}, {-1/2, 1}, {1/2, 1}, {1/2, -1}}];
Ω2 = Rectangle[{-1/2, -1}, {1/2, 1}];

RegionEqual[Ω1, Ω2] returns True, but

mesh1 = ToElementMesh[Ω1, 
   "BoundaryMeshGenerator" -> "Continuation", MaxCellMeasure -> 0.05, 
   "MaxBoundaryCellMeasure" -> 0.05];
mesh2 = ToElementMesh[Ω2, 
   "BoundaryMeshGenerator" -> "Continuation", MaxCellMeasure -> 0.05, 
   "MaxBoundaryCellMeasure" -> 0.05];
Show[RegionPlot[Ω1], mesh1["Wireframe"]]
Show[RegionPlot[Ω2], mesh2["Wireframe"]]

yields two different meshes:

Questions:

Why this behaviour? From the documentation I couldn't find an explanation for this.

How can I control the mesh generation to get a uniform behaviour?

Answer

This is a shortcoming in ToElementMesh. I filed this as a bug, but it will take a bit of time to fix.

As a workaround you could use

mesh1 = 
  ToElementMesh[
    BoundaryDiscretizeRegion[Ω1, 
      MaxCellMeasure -> {"Edges" -> 0.01}], 
    "BoundaryMeshGenerator" -> "Continuation", MaxCellMeasure -> 0.05, 
    "MaxBoundaryCellMeasure" -> 0.01];
Show[RegionPlot[Ω1], mesh1["Wireframe"]]

Comments

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