Full Space Injection preconditioner for SNESVI

[colinjcotter]

I have an idea for a preconditioning approach for SNESVIRS, which might be something I would work on in Dartington.
SNESVIRS is the reduced space Newton method for variational inequalities, which works by dealing with an "active set", which is a subset of the full DOF set of variables where the constraint is (currently) a strict inequality rather than equality.
My proposal is to make a Python preconditioner that injects the active set into the full space, then does something to it with a KSP, and then projects back to the reduced space again (by copying over only the active DOFs).
In particular, the KSP could then do a solve using Auxiliary Operator PC.

I'm interested in comments about whether this is:

1. mathematically sensible,
2. plausible to implement in Firedrake.

[colinjcotter]

People who might be able to comment are:
@wence- , @rckirby , @pbrubeck , @ioannisPApapadopoulos
I'd like to invoke Prof Knepley as well, but cannot.

[pbrubeck]

I don't think that the full space approach is sensible. What about rediscretizing the bilinear form with a DirichletBC with the (iteration-dependent) inactive set?