Numerical experiments with preconditioning by Gram matrix approximation for non-linear elliptic equations

The paper analyzes the numerical performances of the Gram matrix approximat ions preconditioning for solving non-linear elliptic equations. The two tes t problems are non-linear 2-D elliptic equations which describe: (1) the Pl ateau problem and (2) the general pseudohomogeneous model of the catalytic chemical reactor. The standard FEM with piecewise linear test and trial fun ctions is used for discretization. The discrete approximations are solved w ith a double iterative process using the Newton method as outer iteration a nd the preconditioned generalized conjugate gradient methods (CGS and GMRES ) as inner iteration. The Gram matrix approximations consist in replacing t he exact solution of the equation with the preconditioner by few iterations of an appropriate iterative scheme. Two iterative algorithms are tested: i ncomplete Cholesky and multigrid. Numerical experiments indicate that preco nditioners improve the convergence properties of the algorithms for both te st problems. At the second test problem the numerical performances deterior ate at relatively high values of the Pe numbers. (C) 2000 Published by Else vier Science B.V. All rights reserved.