Adaptive inexact Newton methods for discretizations of nonlinear diffusion PDEs. II. Applications

Alexandre Ern (1) , Martin Vohralík (2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 POMDAPI - Environmental Modeling, Optimization and Programming Models

We consider nonlinear algebraic systems resulting 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. In Part I of this work, we have developed a general abstract framework hinging on equilibrated flux reconstructions to derive stopping criteria for both iterative solvers and to control the size and distribution of the overall approximation error. In this Part II, we apply this framework to various discretization schemes like finite elements, nonconforming finite elements, discontinuous Galerkin, finite volumes, and lowest-order mixed finite elements; to different linearizations like fixed point and Newton; and to arbitrary iterative linear solvers. This leads to new guaranteed and robust a posteriori error estimates for nonlinear diffusion problems in the presence of linearization and algebraic errors. Moreover, for many discretization schemes, we improve on, or derive new, flux equilibration techniques.

Mots clés: nonlinear diffusion PDE; nonlinear algebraic system; adaptive linearization; adaptive algebraic solution; a posteriori error estimate; finite elements; nonconforming finite elements; discontinuous Galerkin; finite volumes; mixed finite elements

Domaines: Analyse numérique [math.NA]

Alexandre Ern, Martin Vohralík. Adaptive inexact Newton methods for discretizations of nonlinear diffusion PDEs. II. Applications. 2012. ⟨hal-00681426⟩