To respond to your last comment: are you sure that the v\cdot n value on the boundary also does not decrease with increasing penalty? This may be a bit tricky to visualize in paraview, and may be more appropriately tested by simply integrating the (absolute) quantity in fenics.
It seems that you’re observing some wiggles just a little away from the boundary. This is to be expected, and indeed does not decrease with increasing penalty. Simply put, the discrete function space does not have the flexibility to satisfy all BC’s exactly and something “has to give”. If you force the v\cdot n condition too much, this comes at the cost of a mismatch of the \nabla v condition and/or oscillations in the interior.
If you decrease the penalty parameter, the oscillations may even become a little smaller, or at least the solution will look a little smoother… Of course, there are mathematical conditions on the minimal value the penalty parameter is permitted to have.
Nonlinear problems
For nonlinear problems, Nitsche’s method is indeed a little sketchy. Usually, the consistency term is not an issue (as that simply follows from the usual integration by parts, without performing the substitution of the natural condition), but the symmstry term is; symmetrizing a nonlinear term would imply a nonlinearity on the test function, which is of course not possible.
It seems to me that you can approach this in one of two ways:
- You don’t care too much about the symmetry term. This term exists to make the formulation adjoint consistent, which is required in the Aubin-Nitsche argument for proving optimal convergence in other norms than the natural norm. For non-linear problems, this whole story does not apply anyway. And, in fact, even anti-symmetric variants exist (“Baumann-Oden” in DG, or “non-symmetric Nitsche”).
- You symmetrize in such a way to achieve something else. E.g., to get a symmetric tangent matrix in each Newton iteration, or to make the weak form satisfy some sort of energy condition. In this regard, I dipped my toes in the water in this work: Redirecting.