Preconditioning by approximations of the discrete Laplacian for 2-D non-linear free convection elliptic equations

The paper analyses the preconditioning of non-linear nonsymmetric equations with approximations of the discrete Laplace operator. The test problems ar e non-linear 2-D elliptic equations that describe natural convection, Darcy flow, in a porous medium. The standard second order accurate finite differ ence scheme is used to discretize the models' equations. The discrete appro ximations are solved with a double iterative process using. the Newton meth od as outer iteration and the preconditioned generalised conjugate gradient (PGCG) methods as inner iteration. Three PGCG algorithms, CGN, CGS and GMR ES, are tested The preconditioning with discrete Laplace operator approxima tions consists of replacing the solving of the equation with the preconditi oner by a fete iterations of an appropriate iterative scheme. Two iterative algorithms ave tested: incomplete Cholesky (IC) and multigrid (MG). The nu merical results show that MG preconditioning leads to mesh independence. CG S is the most robust algorithm but its efficiency is lower than that of GMR ES.