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Double Sobolev gradient preconditioning for nonlinear elliptic problems
Author(s) -
Axelsson O.,
Karátson J.
Publication year - 2007
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20207
Subject(s) - mathematics , preconditioner , sobolev space , differentiable function , nonlinear system , mathematical analysis , matrix (chemical analysis) , generalization , linear system , physics , materials science , quantum mechanics , composite material
A mixed variable formulation of a second‐order nonlinear diffusion problem leads to a finite element matrix in a product form. This form enables the efficient updating of the nonlinearity in a Picard type iteration method, in which the preconditioner involves twice a discrete Laplacian. The article gives a conditioning analysis of this method, based on analytic investigations in the corresponding Sobolev function space that reveal the behaviour of this preconditioning. The further generalization of the preconditioner can produce arbitrarily low condition numbers by proper subdivisions of Ω, while still no differentiability of the nonlinear diffusion coefficient is required. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007