Performance of Block-ILU Factorization Preconditioners Based on Block-Size Reduction for 2D Elasticity Systems

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The performance of the block-ILU factorization preconditioners that exploit block-size reduction (introduced in [T. F. Chan and P. S. Vassilevski, Math. Comp., 64 (1995), pp. 129--156]) is studied in the case of block-tridiagonal finite element matrices arising from the discretization of the two-dimensional (2D) Navier equations of elasticity. Conforming triangle finite elements are used for discretizing the differential problem. For the model problem, an estimate of the relative condition number is derived. The efficiency of this incomplete factorization is based on the Sherman--Morrison--Woodbury formula, and, of particular importance, this factorization exists for symmetric and positive definite block-tridiagonal matrices that are not necessarily M-matrices. The convergence rate of the preconditioner is controlled by the block-size reduction parameter which, however, reflects the cost. The numerical tests presented illustrate a strategy for coarse grid size selection; we have also tested the problem for values of the Poisson ratio ($\nu \in (0,\frac{1}{2})$) close to the incompressible limit (e.g., $\tilde \nu =0.9$, where $\nu = \frac{\tilde \nu}{1-\tilde \nu}$) with the expense of about four times more iterations than for the scalar Poisson equation with the same quality preconditioner.