Double Sobolev gradient preconditioning for nonlinear elliptic problems

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 Q, while still no differentiability of the nonlinear diffusion coefficient is required. (c) 2007 Wiley Periodicals, Inc.