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We consider numerical approximations of nonlinear, monotone, and Lipschitz-continuous elliptic problems, with gradient-dependent or gradient-independent diffusivity. For this purpose, we first employ a symmetric iterative linearization, with examples including respectively the Kačanov, Newton, Zarantonello and fixed-point (Picard), L, and M-schemes. We design an iteration-dependent norm, stemming from the concrete linearization and finite element discretization, which leads to an orthogonal decomposition of the total error into the linearization and discretization components. This norm is of reaction-diffusion type, so that available results from a posteriori analysis of linear (singularly perturbed) reaction-diffusion problems enable us to obtain guaranteed and locally efficient error bounds robust with respect to the strength of the nonlinearity. Moreover, the clear distinction and orthogonality of the error components leads to a design of adaptive stopping criteria for iterative linearizations ensuring a natural balance between discretization and linearization components. Under appropriate assumptions, we also show that the iteration-dependent norm is, up to a multiplicative factor tending to one, equivalent to the iteration-independent natural discrete energy norm. In an appendix, we address the Newton (nonsymmetric) linearization for the gradient-independent diffusivity case. Numerical experiments illustrate the theoretical developments.

In this paper, we design a posteriori estimates for finite element approximations of nonlinear elliptic problems satisfying strong-monotonicity and Lipschitz-continuity properties. These estimates include, and build on, any iterative linearization method that satisfies a few clearly identified assumptions; this encompasses the Picard, Newton, and Zarantonello linearizations. The estimates give a guaranteed upper bound on an augmented energy difference (reliability with constant one), as well as a lower bound (efficiency up to a generic constant). We prove that for the Zarantonello linearization, this generic constant only depends on the space dimension, the mesh shape regularity, and possibly the approximation polynomial degree in four or more space dimensions, making the estimates robust with respect to the strength of the nonlinearity. For the other linearizations, there is only a computable dependence on the local variation of the linearization operators. We also derive similar estimates for the energy difference. Numerical experiments illustrate and validate the theoretical results, for both smooth and singular solutions.