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The paper presents the numerical performance of the preconditioned generalized conjugate gradient (PGCG) methods in solving non‐linear convection — diffusion equations. Three non‐linear systems which describe a non‐isothermal chemical reactor, the chemically driven convection in a porous medium and the incompressible steady flow past a sphere are the test problems. The standard second order accurate centred finite difference scheme is used to discretize the models equations. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the PGCG algorithm as inner iteration. Three PGCG techniques, which emerge to be the best performing, are tested. Laplace‐type operators are employed for preconditioning. The results show that the convergence of the PGCG methods depends strongly on the convection—diffusion ratio. The most robust algorithm is GMRES. But even with GMRES non‐convergence occurs when the convection—diffusion ratio exceeds a limit value. This value seems to be influenced by the non‐linearity type.

The paper analyses the preconditioning of non‐linear nonsymmetric equations with approximations of the discrete Laplace operator. The test problems are non‐linear 2‐D elliptic equations that describe natural convection, Darcy flow, in a porous medium. The standard second order accurate finite difference scheme is used to discretize the models’ equations. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the preconditioned generalised conjugate gradient (PGCG) methods as inner iteration. Three PGCG algorithms, CGN, CGS and GMRES, are tested. The preconditioning with discrete Laplace operator approximations consists of replacing the solving of the equation with the preconditioner by a few iterations of an appropriate iterative scheme. Two iterative algorithms are tested: incomplete Cholesky (IC) and multigrid (MG). The numerical results show that MG preconditioning leads to mesh independence. CGS is the most robust algorithm but its efficiency is lower than that of GMRES.