# Can I trust area calculations over discrete MeshRegions? Flipped triangles, possible visual bug

The following code produces a MeshRegion in two different ways. Unfortunately, the first method produces weird artifacts in RegionPlot and Plot3D, whereas the second method causes no such problems. I am hoping to find out if this purely a visual bug or if the mesh is actually corrupted in the first example.

On Mathematica 11.0.1.0 Windows x64 I get the following output:

(* first method *)
<< NDSolveFEM
EM1 = ToElementMesh[Circle[], MaxCellMeasure -> Infinity];
MR1 = MeshRegion[EM1];

(* second method *)
MR2 = DiscretizeRegion[Disk[]];

RegionPlot /@ {MR1, MR2}
Plot3D[-Exp[x^2 + y^2], {x, y} \[Element] #] & /@ {MR1, MR2}


I suspect that this is purely a visual bug because the area of MR1 is calculated very precisely: Area[MR1]/Pi returns 0.996981. Also, FindMeshDefects@MR1 returns no defect. I would hope that someone could shed a light on this weird (visual?) bug, as I am working with discrete MeshRegions like the one from the first method and I am not fully sure if I can trust the area calculations any more.

EDIT: The use of RegionPlot applied to Regions is advertised as a new feature in Mathematica 11 here. The conversion between MeshRegion and ElementMesh is discussed in Section "Comparing ElementMesh and MeshRegion" of this tutorial.

EDIT2: I think the documentation of MeshRegion and/or ToElementMesh need to make the differences between inbuilt regions and discretized regions abundantly clear. This tutorial is a good start but the main documentation is lagging behind. There are many undocumented subtleties that you can only learn from other users. For instance:

• As we see in the present question, RegionPlot struggles with MeshRegions of MeshOrder 2, but does OK with an ElementMesh of the same order: RegionPlot /@ {EM1, MeshRegion[EM1]}. I believe the use of RegionPlot@EM1 is undocumented, so is MeshOrderAlteration, I learned it from user21's answer to the present question.
• Boolean operations such as RegionDifference work for inbuilt regions like Disk[], but not for discretized regions. I know this because C.E. told me so in his answer to this question. think it's not obvious from the documentation that inbuilt and discretized regions are not treated on the same footing. To achieve boolean operations of discretized regions, you need workarounds such as the one demonstrated by Henrik Schumacher in this question.

I hope these problems can be addressed in future versions and the documentation can be updated.

Bug reported to Wolfram CASE:3983893.

• @Szabolcs on V11.2.0/OSX I get the same thing as the OP. EM1 works fine (if I replace MR1 by it). MR1 displays correctly, too, but the RegionPlot of it has the problems in the question. – Michael E2 Oct 9 '17 at 14:28
• @halmir I find it a bit disingenuous to say that "it's not documentation in this precise form so it's not a bug" when it is very clear that something is broken. Also, you are simply wrong that it is not documented. It is used multiple times in the various FEM tutorial pages. – Szabolcs Oct 9 '17 at 14:32
• It is documented. See Section "Comparing ElementMesh and MeshRegion" in reference.wolfram.com/language/FEMDocumentation/tutorial/… – Alexander Erlich Oct 9 '17 at 14:35
• @Szabolcs What I mean is if something is broken by undocumented usage, then it's more like a suggestion than a bug. – halmir Oct 9 '17 at 14:50
• Using regions is now documented for Plot3D; it wasn't in V10 even though the functionality was clearly intended. Currently RegionPlot[reg,...] is still not documented, although examples appear in doc pages other than the documentation page for RegionPlot. Some options have no effect, and one does not have a good idea of the extent and limitations of what can be done. When all else fails, I use the documented DiscretizeRegion instead of RegionPlot. -- That said, the behavior above is clearly unacceptable, and anything so irritating to customers ought to be called a bug. – Michael E2 Oct 9 '17 at 18:54

Here is the correct way to do it:

RegionPlot /@ {EM1, MR2}
Plot3D[-Exp[x^2 + y^2], {x, y} \[Element] #] & /@ {EM1, MR2}


What you are seeing has to do with the fact that ElementMesh use higher order elements (EM1["MeshOrder"]) to approximate the region and functions that take MeshRegion as an argument can not handle that.

To check the area of the ElementMesh use

\[Pi] - Total[EM1["MeshElementMeasure"], 2]
0.009482917945401237


But here is a much more subtle thing to be aware of. If you use the full dimensional region i.e. Disk (like you did in your DiscretizeRegion example - why the inconsistency?) not Circle you get a much better representation:

EM2 = ToElementMesh[Disk[], MaxCellMeasure -> Infinity];
\[Pi] - Total[EM2["MeshElementMeasure"], 2]
2.2342021526178257*^-6


It may be possible to improve the Circle case but it's not a high priority.

Another thing you can do is convert the higher order ElementMesh to a first order mesh and then do the conversion like so:

RegionPlot[MeshRegion[MeshOrderAlteration[EM2, 1]]]


A few things to know: An ElementMesh is not a Region as you can check that with RegionQ. This may change in the future but that's the situation in V11.2. The primary purpose in ElementMesh 's life is to serve as a mesh for the Finite Element Method and have a default mesh order of of 2. As such the ElementMesh is used in NDSolve and related functions such as InterpolatingFunction objects. It can also be used in NIntegrate. I think all of these usages are shown in one of the FEM tutorials. An ElementMesh can be converted to a MeshRegion and the resulting MeshRegion is also second order accurate. This is undocumented MeshRegion functionality but needed to make the geometry frame work to work. A conversion to a second order MeshRegions does not mean, however, that all functions (that allow RegionQ) can deal with that (yet), which is understandable, I think. But does that mean an ElementMesh should also be first order accurate only? Certainly not - that would give lousy FEM results. So what you see here is issues/bugs as the arise at the interface of projects. I filed the plotting over second order mesh regions as a bug; that said I can not say when it's fixed as this is not my department and I have little control over that. The documentation you cite as 'a good start' is one from the FEM framework that I wrote. If you have suggestions for that I am open to changes etc; but I am not going to document short comings of MeshRegion there - it's just not the right place. So my suggestion is that you report MeshRegion issues to wolfram. It does make a difference if issues are reported in house or by customers. Having said that I added an example for MeshOrderAlteration to the ElementMesh generation tutorial, pending approval. The fact that you can plot over ElementMesh is made use of several times in that tutorial - granted it does not show it for every single plot... So, again, if you have anything that is ElementMesh oriented and you'd like to see documented or where the documentation should be improved let me know.
• Thank you, this is exactly the kind of answer I was hoping for. The fact that RegionPlot is made for MeshOrder 1 explains everything. A few remarks: 1) Plot3D[-Exp[x^2 + y^2], {x, y} \[Element] EM1] returns error message on my machine: Plot3D::idomdim: {x,y}\[Element]EM1 does not have a valid dimension as a plotting domain.. 2) Yet another undocumented feature (AFAIK) is that you can integrate over an ElementMesh: For instance, NIntegrate[ Exp[x^2 + y^2], {x, y} \[Element] #] & /@ {MeshOrderAlteration[ EM1, 1], EM1 } // Differences works. – Alexander Erlich Oct 10 '17 at 13:06
• Just as a personal note, this is now my third question related to discrete MeshRegions (the other two are here and here). Pre-programmed regions like Disk[] and discrete ones (coming from ToElementMesh or imported from STL) are really not on the same footing in MMA. I think the documentation of e.g. MeshRegion should highlight that more. – Alexander Erlich Oct 10 '17 at 13:19
• @AlexanderErlich if you have suggestions for how the FEM documentation or functionality should improve (e.g. ToElementMesh), I am all ears and I'll try to make it happen. For other things (like MeshRegion, DiscretizeRegion) it's probably better to contact support and send in your suggestions. – user21 Oct 10 '17 at 14:51
• @AlexanderErlich, the Plot3D error is fixed in V11.2. – user21 Oct 10 '17 at 22:02