We consider nonlinear algebraic systems arising from numerical
discretizations of nonlinear partial differential equations of
diffusion type. In order to solve them, some iterative nonlinear
solver, and, on each step of this solver, some iterative linear solver
are used. We propose an adaptive choice of the number of steps of both
the linear and nonlinear solvers. Both stopping criteria are based on
an a posteriori error estimate which distinguishes the different error
components, namely the algebraic error, the linearization error, and
the discretization error; we stop whenever the corresponding error
does not affect the overall error significantly. Our estimates also
give a guaranteed upper bound on the overall error at each step of the
nonlinear and linear solvers. We prove the (local) efficiency and
robustness of our estimates with respect to the size of the
nonlinearity. This is achieved thanks to the choice of the error
measure, the dual norm of the residual augmented by a jump seminorm.
Our developments are carried at an abstract level, yielding a general
framework. We apply this framework to the fixed point and Newton
linearizations and to most common discretization methods: the finite
element, the nonconforming finite element, the discontinuous Galerkin,
and finite volume and mixed finite element ones. All iterative linear
solvers are covered. Numerical experiments illustrate the tight
overall error control and important computational savings achieved by
our approach.