refactor: splitting up of NonlinearSolve #126

Job	Run time
downgrade (1.10, core)	2m 13s
downgrade (1.10, adjoint)	2m 12s
test (1.10, ubuntu-latest, core)	2m 12s
downgrade (1.10, alloc_check)	2m 14s
test (1.10, ubuntu-latest, adjoint)	2m 12s
test (1.10, ubuntu-latest, alloc_check)	2m 12s
test (1.10, macos-latest, core)	1s
test (1.10, macos-latest, adjoint)	1s
test (1.10, macos-latest, alloc_check)	1s
test (1.10, windows-latest, core)	2m 6s
test (1.10, windows-latest, adjoint)	2m 7s
test (1.10, windows-latest, alloc_check)	2m 5s
test (1, ubuntu-latest, core)	2m 6s
test (1, ubuntu-latest, adjoint)	2m 7s
test (1, ubuntu-latest, alloc_check)	2m 4s
test (1, macos-latest, core)	1s
test (1, macos-latest, adjoint)	1s
test (1, macos-latest, alloc_check)	1s
test (1, windows-latest, core)	2m 4s
test (1, windows-latest, adjoint)	2m 7s
test (1, windows-latest, alloc_check)	2m 9s
	32m 16s

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