Guaranteed, locally efficient, and robust a posteriori estimates for nonlinear elliptic problems in iteration-dependent norms. An orthogonal decomposition result based on iterative linearization - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Preprints, Working Papers, ... Year : 2023

Guaranteed, locally efficient, and robust a posteriori estimates for nonlinear elliptic problems in iteration-dependent norms. An orthogonal decomposition result based on iterative linearization

Abstract

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.
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hal-04156711 , version 1 (09-07-2023)

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  • HAL Id : hal-04156711 , version 1

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Koondanibha Mitra, Martin Vohralík. Guaranteed, locally efficient, and robust a posteriori estimates for nonlinear elliptic problems in iteration-dependent norms. An orthogonal decomposition result based on iterative linearization. 2023. ⟨hal-04156711⟩
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