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I was trying to get this done, but in the current state PETSc isn't even installable with DiffEqBase:
(NonlinearSolve) pkg> add PETSc
Resolving package versions...
ERROR: Unsatisfiable requirements detected for package DocStringExtensions [ffbed154]:
DocStringExtensions [ffbed154] log:
├─possible versions are:0.4.6-0.9.3 or uninstalled
├─restricted by compatibility requirements with DiffEqBase [2b5f629d] to versions:0.9.0-0.9.3
│ └─DiffEqBase [2b5f629d] log:
│ ├─possible versions are:3.13.2-6.147.0 or uninstalled
│ └─restricted to versions 6.146.0-6 by NonlinearSolve [8913a72c], leaving only versions:6.146.0-6.147.0
│ └─NonlinearSolve [8913a72c] log:
│ ├─possible versions are:3.5.6 or uninstalled
│ └─NonlinearSolve [8913a72c] is fixed to version 3.5.6
└─restricted by compatibility requirements with MPI [da04e1cc] to versions:0.8.0-0.8.6 — no versions left
└─MPI [da04e1cc] log:
├─possible versions are:0.7.0-0.20.19 or uninstalled
└─restricted by compatibility requirements with PETSc [ace2c81b] to versions:0.15.0-0.19.2
└─PETSc [ace2c81b] log:
├─possible versions are:0.1.2-0.1.3 or uninstalled
└─restricted to versions * by an explicit requirement, leaving only versions:0.1.2-0.1.3
Also, their Julia documentation is extremely sparse which makes it almost impossible to figure out how to use the different solvers.
There is https://github.com/gridap/GridapPETSc.jl/ but again the amount of docs they have is pretty much zero apart from API (unless I am looking at the wrong place).
I also seem to remember there was a recent package which wrapped the Petsc functionality in julia but lacked support for the nonlinear stuff.
Point being I don't think this will be handled anytime soon, till we have a decent Petsc wrapper in Julia.
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