How to find values between nodes in finite element

the best way to find the stress at a point is to point a node there, ie include the point in the mesh.

the next best is to interpolate with the data you have. with 3 co-ordinates you can solve three equations (simply) ... ie pick three nodes and their stress, and solve the three equations for the stress at the chosen point. you can do this several times with different points and average, or you can solve the over-defined problem (using many points so you have more equations than unknowns).

another day in paradise, or is paradise one day closer ?

I've developed a short work on this, take it as a mini-blog, diving into the mechanics of plate breakdown.
The heartiest solution is the static condensation, 'aka guyan technique'
.

@fugeeo:

Unless I am missing something, your subdivided meshes are not "compatible" (as you have subdivided nodes on the finer subdivided elements which are not connected to any corresponding nodes on the adjoining un-subdivided element), and this will be yielding spurious results. This accounts for the irregular stress contour plots - you should be able to generate nice smooth / regular contour plots if you use compatible graded meshes. (In effect, you have modelled slits in your plate along the subdivided edges, with stress concentrations at the nodes where they reconnect.)

You are not missing anything, there are free edges in the mesh.

Aerospace Stress Analysis and FEA Courses
Stressing Stresslessly!

personally, i have no idea what fugeeo's post is meant to mean ...

another day in paradise, or is paradise one day closer ?

me either but those sure are pretty images which is all that matters