Istituto di Matematica Applicata e Tecnologie Informatiche     
Fassino C., Manzini G. Iterative methods for the non-linear Richards' equation. Preprint ercim.cnr.ian//1997-1073, 1997.
A mixed-hybrid formulation of the non-linear Richards' equation based on the discontinuous lowest order Raviart-Thomas elements produces a non-linear algebraic problem, whose solution demands for an appropriate non-linear iterative method. In this work we investigate and compare the performance of the Newton, (relaxed) Picard and fast quasi-Newton linearizations when applied to 1-D and 2-D stationary and evolutionary model problems. The Newton method, although only locally convergent, results to be the most effective resolution strategy, whilst the Picard method, has only a linear convergence rate, which may be too slow for practical applications. Broyden-type methods are also locally but superlinearly convergent, and they do not require the calculation of the Jacobian matrix as for the Newton method. An acceleration technique of the Picard iterative scheme, based on a fast Broyden-type linearization, is thus proposed. The method is particularly promising for its super-linear convergence rate, its computational efficiency, and its simplicity of implementation.
Subject Richards' equation, nonlinear iterative methods, mixed hybrid FEM

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