refactor: Move dual nonlinear solving to NonlinearSolveBase #194

Job	Run time
downgrade (1.10, core)	11m 43s
downgrade (1.10, adjoint)	9m 0s
downgrade (1.10, alloc_check)	9m 30s
test (1.10, ubuntu-latest, core)	11m 58s
test (1.10, ubuntu-latest, adjoint)	9m 21s
test (1.10, ubuntu-latest, alloc_check)	9m 7s
test (1.10, macos-latest, core)	6m 47s
test (1.10, macos-latest, adjoint)	6m 31s
test (1.10, macos-latest, alloc_check)	5m 39s
test (1.10, windows-latest, core)	12m 39s
test (1.10, windows-latest, adjoint)	10m 19s
test (1.10, windows-latest, alloc_check)	10m 55s
test (1, ubuntu-latest, core)	20m 10s
test (1, ubuntu-latest, adjoint)	15m 44s
test (1, ubuntu-latest, alloc_check)	14m 52s
test (1, macos-latest, core)	7m 3s
test (1, macos-latest, adjoint)	10m 30s
test (1, macos-latest, alloc_check)	9m 23s
test (1, windows-latest, core)	18m 45s
test (1, windows-latest, adjoint)	17m 1s
test (1, windows-latest, alloc_check)	16m 31s
	4h 3m 28s

Provide feedback