Fast solution to finite element flow equations by Newton iteration and modified conjugate gradient method
Abstract
The modified conjugate gradient method with incomplete factorization coupled with an initial Newton iterative scheme is applied to the solution of finite element equations arising from the variational formulation of boundary value problems of ground water flow. An irregular triangular mesh is used for a twolayer aquifer, and a more complex example involving a ninelayer aquitardaquifer system is also considered. It is found that, after a number of Newton iterations that do not reduce significantly the initial error but instead are used to eliminate a large number of residual components in the space of the eigenvectors of the iteration matrix, the modified conjugate gradient method with the incomplete Cholesky decomposition achieves the exact solution in few steps with a great improvement in accuracy. Depending on the desired final accuracy, up to 50% of the modified conjugate gradient iterations may be replaced with Newton iterations at a significant savings in computational time. It is also pointed out that the present method is more useful than the successive overrelaxation technique with optimum overrelaxation factor and the firstdegree Chebyshev iteration.
 Publication:

International Journal for Numerical Methods in Engineering
 Pub Date:
 May 1980
 DOI:
 10.1002/nme.1620150504
 Bibcode:
 1980IJNME..15..661G
 Keywords:

 Boundary Value Problems;
 Conjugates;
 Finite Element Method;
 Flow Equations;
 Iteration;
 Newton Methods;
 Water Flow;
 Aquifers;
 Chebyshev Approximation;
 Eigenvectors;
 Ground Water;
 Relaxation Method (Mathematics);
 Variational Principles;
 Fluid Mechanics and Heat Transfer