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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?

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  • $\begingroup$ There is a fairly simple comparison of Sobolev norms where the problem has a weak formulation and the discrete solution follows by using a Galerkin method. I suspect that you know as much, but little more can be said without discussing why the problem lacks a natural space and what degree of regularity the boundary possesses. $\endgroup$ – hardmath Apr 18 at 2:11

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