In contrast to the case with two concentric Balls which generate the errors when executing ToElementMesh on the RegionDifference, using Cuboids does not generate the error:

Needs["NDSolve`FEM`"]; 

inner = 10;
outer = 20;

c1 = Cuboid[{-inner, -inner, -inner}, {inner, inner, inner}];

c2 = Cuboid[{-outer, -outer, -outer}, {outer, outer, outer}];

\[CapitalOmega] = RegionDifference[c2, c1];

RegionPlot3D[\[CapitalOmega], PlotStyle -> Opacity[.5]] 

mesh = ToElementMesh[\[CapitalOmega], MaxCellMeasure -> {"Volume" -> 1}]; 

The generation of the mesh is time consuming and especially so as the "Volume" is decreased. One can see that the number of mesh elements in this case is already quite significant:

In[54]:= mesh

Out[54]= ElementMesh[{{-20., 20.}, {-20., 20.}, {-20., 
   20.}}, {TetrahedronElement["<" 106250 ">"]}]

All of this suggests a bug in the Ball case....

I also encountered Udo's bug (http://community.wolfram.com/groups/-/m/t/357364) when exploring plots of the above RegionPlot3D.

Dear Dr. Krause,

Thanks a great deal for your reaction.

To explain better what I did and why I did it: I am a relative newcomer to Mathematica and this is the first time I tried to use the 3D-PDE module of Mathematica. I tried my hand on this problem because: 1. the domain is finite so that it lends itself to a Finite Element treatment; 2. the solution is a very simple analytical expression. It's nice to have a check on the Mathematica outcome.

To stay on the safe side I tried - as much as possible - to copy code I read in the Mathematica documentation on this subject (http://reference.wolfram.com/language/FEMDocumentation/tutorial/SolvingPDEwithFEM.html). I agree with you that there's nothing implicit in my first line of code. I also tried the region definition without the ImplicitRegion statement; e.g. as a RegionDifference between a Ball of radius 100 and a Ball of radius 1. That didn't help, unfortunately: I got the same error message.

I will try out your suggestion to do everything consistently in Cartesian coordinates although it "feels" very weird to do that in a geometry that so obviously has spherical symmetry. I will let you know whether that works.

Thanks, René Samson

It has to do with the ToElementMesh[] function

In[1]:= Needs["NDSolve`FEM`"]

In[2]:= Clear[\[CapitalOmega]]
\[CapitalOmega] = ImplicitRegion[2 <= (x^2 + y^2 + z^2) <= 4, {x, y, z}]

Out[3]= ImplicitRegion[2 <= x^2 + y^2 + z^2 <= 4, {x, y, z}]

In[4]:= mesh = ToElementMesh[\[CapitalOmega], MaxCellMeasure -> 0.01];
During evaluation of In[4]:= ToElementMesh::fememib: The input has or generated an intersecting boundary and cannot be processed. >>
During evaluation of In[4]:= ToElementMesh::femtemnm: A mesh could not be generated. >>

this function has interesting options,

In[5]:= Options[ToElementMesh]

Out[5]= {"BoundaryMarkerFunction" -> None, 
 "BoundaryMeshGenerator" -> Automatic, 
 "CheckIncidentsCompletness" -> True, "CheckIntersections" -> True, 
 "CheckQuality" -> Automatic, "DeleteDuplicateCoordinates" -> True, 
 "ElementMeshGenerator" -> Automatic, 
 "ImproveBoundaryPosition" -> Automatic, "IncludePoints" -> {}, 
 "MaxBoundaryCellMeasure" -> Automatic, "MeshElementBlocks" -> 1, 
 "MeshElementConstraint" -> Automatic, "MeshElementType" -> Automatic,
  "MeshOrder" -> Automatic, "MessageHead" -> Automatic, 
 "NodeReordering" -> Automatic, "PointMarkerFunction" -> None, 
 "RegionHoles" -> Automatic, "RegionMarker" -> None, 
 "SteinerPoints" -> Automatic, AccuracyGoal -> Automatic, 
 MaxCellMeasure -> Automatic, MeshQualityGoal -> Automatic, 
 MeshRefinementFunction -> None, PrecisionGoal -> Automatic, 
 PerformanceGoal :> $PerformanceGoal}

some of them, like RegionHoles are still undocumented. Nevertheless, the solid ball works in 3D out of the box

only the result has to be understood.