| We consider nonlinear algebraic systems resulting from numerical discretizations of nonlinear partial differential equations of diffusion type. To solve these systems, some iterative nonlinear solver, and, on each step of this solver, some iterative linear solver are used. In this first part, we derive adaptive stopping criteria for both iterative solvers. Both criteria are based on an a posteriori error estimate which distinguishes the different error components, namely the discretization error, the linearization error, and the algebraic error. We stop the iterations whenever the corresponding error does no longer affect the overall error significantly. Our estimates also yield 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 the estimates with respect to the size of the nonlinearity owing, in particular, to the error measure involving the dual norm of the residual. Our developments are carried at an abstract level, yielding a general framework. We show how to apply this framework to the Crouzeix--Raviart nonconforming finite element discretization, Newton linearization, and conjugate gradient algebraic solution, and we illustrate on numerical experiments for the $p$-Laplacian the tight overall error control and important computational savings achieved in our approach. Part II is devoted to the application of our abstract framework to a broad class of discretization methods. |
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