G. Juncu et C. Popa, Numerical experiments with preconditioning by Gram matrix approximation for non-linear elliptic equations, MATH COMP S, 52(1), 2000, pp. 53-71
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.