refactor: Move dual nonlinear solving to NonlinearSolveBase #197

Job	Run time
downgrade (1.10, core)	12m 12s
downgrade (1.10, adjoint)	9m 4s
test (1.10, ubuntu-latest, core)	11m 56s
downgrade (1.10, alloc_check)	9m 3s
test (1.10, ubuntu-latest, adjoint)	9m 21s
test (1.10, ubuntu-latest, alloc_check)	9m 45s
test (1.10, macos-latest, core)	6m 59s
test (1.10, macos-latest, adjoint)	6m 49s
test (1.10, macos-latest, alloc_check)	6m 58s
test (1.10, windows-latest, core)	12m 15s
test (1.10, windows-latest, adjoint)	10m 44s
test (1.10, windows-latest, alloc_check)	10m 41s
test (1, ubuntu-latest, core)	20m 17s
test (1, ubuntu-latest, adjoint)	15m 26s
test (1, ubuntu-latest, alloc_check)	15m 11s
test (1, macos-latest, core)	8m 50s
test (1, macos-latest, adjoint)	9m 50s
test (1, macos-latest, alloc_check)	11m 59s
test (1, windows-latest, core)	18m 47s
test (1, windows-latest, adjoint)	17m 24s
test (1, windows-latest, alloc_check)	16m 45s
	4h 10m 16s

Provide feedback