Fast solution to finite element flow equations by newton iteration and modified conjugate gradient method

Abstract

AbstractA modification (Irons;6 Concus et al. 1) to the conjugate gradient (CG) method by Hestenes and Stiefel5 has recently renewed the interest in this elegant technique which appears to be extraordinarily promising for large sparse systems Ax = b, where A is a symmetric positive definite matrix. A good approximation K−1 for the inverse of A is needed in the modified algorithm. For finite difference sets of equations, Meijerink and van der Vorst9 and Kershaw7 have experienced a very fast convergence with a matrix K−1 determined by the incomplete Cholesky decomposition of A.A further acceleration of the iteration may be achieved by preliminarily processing the initial guessed solution by the Newton iterative scheme before using the modified conjugate gradient (MCG) method. The initial Newton iterations (NI) have the useful property of significantly reducing the components of the residual r0 along the eigenvectors of AK−1 associated with the eigenvalues lying in the vicinity of 1. The latter are expected to include the vast majority of the eigenvalues of AK−1. As a result the MCG method is left with a smaller number of r0 components to set to zero in a reduced dimensional eigenvector space and hence the solution is arrived at in fewer iterations. Depending on the desired final accuracy, up to 50 per cent of the MCG iterations may be equivalently replaced by an equal number of NI which are computationally faster.This approach has been applied to the solution of finite element sets of linear equations arising from the arbitrarily irregular triangle discretization of groundwater flow domains in both steady and unsteady conditions. For diagonally dominant matrices the results emphasize the excellent performance of the MCG method which proved much faster than the first‐degree Chebyshev iteration (CHI) and required a number of iterations an order of magnitude smaller than the successive over‐relaxation technique (SOR) with optimum over‐relaxation factor.