To Export the full information contained in ToElementMesh you have to convert it into a list, then transform all the possible exponentials to plain decimals so that you can export the whole list as a string without notation issues.
mesh = ToElementMesh[Disk[]]
Export["diskmesh.txt",NumberForm[#, ExponentFunction -> (Null &)] & /@ List @@ mesh, "String"]
Now, note that the mesh you created has four elements
In[175]:= Length@mesh
Out[175]= 4
These are "Coordinates", "MeshElements", "BoundaryElements", "PointElements"
Therefore, when you recreate ToElementMesh from the imported .txt file you gotta specify what is what, like this
In[177]:= #1->#2 & ~MapThread~{{"Coordinates", "MeshElements","BoundaryElements","PointElements"},ToExpression@Import["diskmesh.txt"]}//ToElementMesh@@#&
Out[177]= ElementMesh[{{-1., 1.}, {-1., 1.}}, {TriangleElement[<504>]}]
em["Coordinates"])
, then useExport
with the"Table"
format. But the coordinates don't contain full information about the mesh and in the original version of the question you asked about exporting the "mesh", not just the "coordinates". $\endgroup$