Experimental comparison of three-dimensional point and line modified incomplete factorizations

We examine how the variations of the coefficients of 3-dimensional (3D) partial differential equations (PDEs) influence the convergence of the conjugate gradient method, preconditioned by standard pointwise and linewise modified incomplete factorizations. General analytical spectral bounds obtained previously are applied, which displays the conditions under which good performances could be expected. The arguments also reveal that, if the total number of unknowns is very large or the number of unknowns in one direction is much larger than in both other ones, or if there are strong jumps in the variation of the PDE coefficients or fewer Dirichlet boundary conditions, then linewise preconditionings could be significantly more efficient than the corresponding pointwise ones. We also discuss reasons to explain why in the case of constant PDE coefficients, the advantage of preferring linewise methods to pointwise ones is not as pronounced as in 2D problems. Results of numerical experiments are reported.

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