1. ## steady state solution and element number

dear dr.sewell,
I am using my pc version 0f pde2d to solve a simple steady state problem of heat 2d diffusion where conductivity is function of temperature and on the top of the layer there is a very thin layer with a very low conductivity.
The problem is that using "adapt" and a growing number of elements I never found the same solution being the temperature field always increasing. May you help me? The pde2d
file is already sent to you.
thanks a lot

costanzo federico

2. Constanzo,
Your nonlinear conductivity coefficient has a discontinuity at y=aH-10,
I added an extra grid line:
YGRID(NYGRID/2) = aH-10
and now it seems to work fine (also increased NYGRID to 21, but that's not
critical).

The solution has a discontinuity in its derivatives along this line, so it is
critical that no initial triangles, and thus no final triangles, straddle this
interface. Remember that the approximate solution is very smooth
(polynomial) everywhere except at the triangle boundaries, where it also
has discontinuous first derivatives, so you get a MUCH better approximation
of you match the discontinuity (in derivatives) of the approximate solution
and the discontinuity of the true solution. You can imagine that it is hard
to accurately approximate a solution with a discontinuous derivatives by
a smooth polynomial.

In fact it is so critical to avoid triangles which straddle such interfaces, that
if you have a curved interface, PDE2D will follow the curved interface when
refining the triangulation, even using isoparametric triangles (triangles with
curved edges) along the discontinuity.

With the new grid line, the solution is almost unchanged when you rerun
with a different adaptive triangulation. This accurate solution has a temp
of about 450 at the bottom, your previous runs produced solutions which
changed each time you ran with a better adaptive triangulation, and
appeared to be approaching this value from below, but very slowly.