Could not generate mesh when the mesh count is too large

Question

I am trying to generate 1D mesh points on x axis, shown below:

Needs["NDSolve`FEM`"]
eC = 100(*Element Count*);
xMesh = ToGradedMesh[{{Line[{{0}, {3}}], <|"Alignment" -> "Right", 
      "ElementCount" -> eC|>}, {Line[{{3}, {4}}], <|
      "Alignment" -> "Left", 
      "ElementCount" -> eC|>}, {Line[{{4}, {5}}], <|
      "Alignment" -> "Right", "ElementCount" -> eC, 
      "GradingRatio" -> 1.2|>}, {Line[{{5}, {7}}], <|
      "Alignment" -> "Left", "ElementCount" -> eC, 
      "GradingRatio" -> 1.2|>}}];
MeshRegion[xMesh]

The code worked fine when my element count eC is 10. However, when I try to increase the element count eC to 100, the following error arises:

For more context, I want the mesh points to concentrate in certain regions (x=3, and x=5). Therefore, I used GradingRatio to tune the non-uniformity of their distributions. I realized that the error above also tends to occur when the GradingRatio value increases.

user21 · Accepted Answer · 2025-06-23 12:28:55Z

The exact reasoning behind this is somewhat complicated. But I'll show a way to modify this; even when I do not advise to do it. Let's look at a simpler example:

Needs["NDSolve`FEM`"]
eC = 80;(* workds *)
eC = 81(*fails*);
xMesh = ToGradedMesh[{
   {Line[{{4}, {5}}], <|"Alignment" -> "Right", "ElementCount" -> eC, 
     "GradingRatio" -> 1.2|>}}]

This fails to generate a mesh. using 80 elements will work. Once you have called ToGradedMesh once you have access to an internal variable

NDSolve`FEM`ToGradedMeshDump`$elementTolerance // N
(* 1.*10^-8 *)

If you relax this safety factor a bit, to say:

NDSolve`FEM`ToGradedMeshDump`$elementTolerance = 10^-6;

Then this will work:

eC = 100;
xMesh = ToGradedMesh[{
   {Line[{{4}, {5}}], <|"Alignment" -> "Right", "ElementCount" -> eC, 
     "GradingRatio" -> 1.2|>}}]

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