>>> I'm not an expert on Libmesh, but I recall that PETSC allows one to
>>> set the iinitial guess. If PETSc can be accessed through the libmesh
>>> interface then it should be possible quite easily.
>> 
>> Indeed it is -- this is in fact the default behavior in libMesh.  Whatever
>> is in the System.solution vector is handed off to the linear solver as the
>> initial guess.  You should be able to verify this by calling
>> 
>> equation_systems.solve();
>> equation_systems.solve();
>> 
>> And seeing that the second solve converges in 0 iterations.
> ...
> 
> It seems there is no change...
> 

I'm not sure what is going on in your application...  If I take ex4 and add
a second solve as mentioned above and run

$ ./ex4-dbg -d 2 -n 20  -ksp_converged_reason

I get

Linear solve converged due to CONVERGED_RTOL iterations 78
Linear solve converged due to CONVERGED_RTOL iterations 0


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