I am solving a boundary value problem (BVP) that involves a system of equations (similar to the Euler or Navier-Stokes equations) for which, at this moment, there exists no sufficient theory to define their natural space.
I am interested in the restriction of the BVP solution on the boundary. So, my question is:
Can I get insight into a -possibly- natural space by computing the discrete Sobolev norms of the restricted solution on the boundary?
By this I mean that I will be progressively refining the mesh and checking in which norms the solution converges.
Is there any theory regarding discrete equivalents of Sobolev norms and is that even a practice one can follow to obtain some insight?