refactor: splitting up of NonlinearSolve #123

Job	Run time
downgrade (1.10, core)	13m 7s
downgrade (1.10, adjoint)	8m 36s
downgrade (1.10, alloc_check)	8m 23s
test (1.10, ubuntu-latest, core)	12m 53s
test (1.10, ubuntu-latest, adjoint)	8m 23s
test (1.10, ubuntu-latest, alloc_check)	8m 38s
test (1.10, macos-latest, core)	12m 7s
test (1.10, macos-latest, adjoint)	6m 41s
test (1.10, macos-latest, alloc_check)	6m 6s
test (1.10, windows-latest, core)	14m 7s
test (1.10, windows-latest, adjoint)	10m 22s
test (1.10, windows-latest, alloc_check)	10m 1s
test (1, ubuntu-latest, core)	22m 21s
test (1, ubuntu-latest, adjoint)	15m 24s
test (1, ubuntu-latest, alloc_check)	14m 30s
test (1, macos-latest, core)	17m 35s
test (1, macos-latest, adjoint)	8m 49s
test (1, macos-latest, alloc_check)	11m 47s
test (1, windows-latest, core)	22m 41s
test (1, windows-latest, adjoint)	16m 59s
test (1, windows-latest, alloc_check)	16m 44s
	4h 26m 14s

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