We consider here two classes of non-linear systems giving different degrees of non-linearity. In both cases the systems arise from finite difference discretisations of non-linear elliptic partial differential equations. Our solution methods can also fit into two categories - linearisation and non-linearisation techniques - and in our studies we have pursued three main objectives. 1) For mildly non-linear systems we generalise certain iterative techniques from the solution of linear systems to the solution of nonlinear systems with symmetric Jacobians. We are especially concerned with the effect of preconditioning of the equations here. 2) We consider the use of bidiagonalisation on non-linear systems, using preconditioning in two ways and considering both classes of nonlinear problem. 3) We solve the laminar flow problem and assess the effects of multigrid acceleration on non-linear S.I.P. techniques.
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