The numerical solution of elliptic partial differential equations (PDEs) leads to the solution of systems of linear algebraic equations of the form Ax=b. The solution of such linear systems is very time consuming, especially if three-dimensional (3D) elliptic PDEs are to be handled by using fine grid discretization. The application of direct methods is practically impossible in the latter case. Therefore, the large systems of linear algebraic equations arising after the discretization of 3D elliptic PDEs have to be treated using different iterative methods.

In 1979, Björck and Elfving proposed an iterative method based on the well-known symmetric successive over-relaxation (SSOR) algorithm, accelerated by the conjugate gradient (CG) algorithm, and applied to the system of normal equations (obtained by multiplying Ax=b by the transpose of matrix A) [1]. This algorithm is called CGMN.

Gordon and Gordon introduce an extended version of CGMN to allow the use of cyclic relaxation parameters, called CGMN cyclic (CGMNC). They also define different related algorithms that are used in the comparisons. Then, they list nine test examples--containing the Laplace operator as well as some convection terms and right-hand sides--that are used in the numerical tests. They select the incomplete lower-upper (LU) factorization with thresholding (ILUT) preconditioner proposed by Saad in 2003 [2], and apply it with default parameters (drop tolerance=0 and fill-in=1.0). They apply stopping criteria based on the application of relative residual norms. The linear systems are scaled before the start of the iterative process. The authors present a comprehensive set of numerical results and carefully analyze the performance of the different algorithms.

The numerical experiments indicate that the CGMC algorithm performs very well for all test examples used in this paper.