I would like to thank you for all the suggestions for the problem with the heat transportation equation I have posted time ago! I rewrited the code along with the implementation of the Neumann boundary condition, changed the discretisation and eliminated questionable boundary-condition that could violate the conservation of energy in the following way:
T = NDSolveValue[{-Laplacian[t[r, \[Theta], z], {r, \[Theta], z},
"Cylindrical"]*(10^-5) +
0.001*(1 - (r^2))*D[t[r, \[Theta], z], z] ==
NeumannValue[0, r == 10^-7], DirichletCondition[t[r, \[Theta], z] == 10,
z == 10^-7 && 10^-7 <= r <= 1 && 0 <= \[Theta] <= 2*\[Pi]], DirichletCondition[t[r, \[Theta], z] == 0,
10^-7 <= z <= 10 && r == 1 && 0 <= \[Theta] <= 2*\[Pi]]}, t, {r, 10^-7, 1}, {\[Theta], 10^-7, 2*\[Pi]}, {z, 10^-7, 10}, MaxStepSize -> 0.0001]
and used the command:
Plot3D[T[r, \[Theta], z] /. \[Theta] -> \[Pi], {r, 0, 1}, {z, 0, 1}, PlotRange -> All]
or
ContourPlot[T[r, \[Theta], z] /. \[Theta] -> \[Pi]/2, {r, 0, 1}, {z, 0, 1},MaxRecursion -> 2]
to plot the solution
Now is almost the solution it needs to be, but as you can see from the 3D representation, I keep getting that small bump at the beginning and also a fluctuation at the edge that brings the temperature below 0 which is obviously wrong.I don' t know any other command that could be implemented in the code to make it work better, because it seems that the change in discretization and the implementation of the NeumallValue are not enoug to ensure the right solution in all points.