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

The paper analyzes the numerical performances of the Gram matrix approximations preconditioning for solving non-linear elliptic equations. The two test problems are non-linear 2-D elliptic equations which describe: (1) the Plateau problem and (2) the general pseudohomogeneous model of the catalytic chemical reactor. The standard FEM with piecewise linear test and trial functions is used for discretization. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the preconditioned generalized conjugate gradient methods (CGS and GMRES) as inner iteration. The Gram matrix approximations consist in replacing the exact solution of the equation with the preconditioner by few iterations of an appropriate iterative scheme. Two iterative algorithms are tested: incomplete Cholesky and multigrid. Numerical experiments indicate that preconditioners improve the convergence properties of the algorithms for both test problems. At the second test problem the numerical performances deteriorate at relatively high values of the Pe numbers.