## A new mixed preconditioning method for finite element computations.(English)Zbl 0762.65060

The element-by-element preconditioning method is generalized in two directions in order to achieve better convergence properties when applied in connection with conjugate gradients or GMRES. In a first step the set of elements is partitioned into clusters of elements so that the global stiffness matrix $$A$$ can be written as a sum of matrices corresponding to the clusters.
On the base of this representation the clustered element-by-element preconditioner is defined in a similar way by a sequential product of cluster level matrices. In a second step a cluster companion preconditioning is defined based on a companion mesh that has some analogy to multigrid.
However, the companion preconditioner requires a regularization to become positive definite. The two preconditioners are finally mixed in order to exploit the coupling properties of both in an optimal way. Numerical tests illustrate the gain of convergence rate.

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text:

### References:

 [1] Hughes, T.J.R.; Levit, I.; Winget, J., An element-by-element solution algorithm for problems of structural and solid mechanics, Comput. methods appl. mech. engrg., 36, 241-254, (1983) · Zbl 0487.73083 [2] Hughes, T.J.R.; Winget, J.; Levit, I.; Tezduyar, T.E., New alternating direction procedures in finite element analysis based upon EBB approximate factorizations, (), 75-109 · Zbl 0563.73052 [3] Hughes, T.J.R.; Ferencz, R.M., Fully vectorized EBE preconditioners for nonlinear solid mechanics: applications to large-scale three-dimensional continuum, shell and contact/impact problems, (), 261-280 [4] Tezduyar, T.E.; Liou, J., Element-by-element and implicit-explicit finite element formulations in computational fluid dynamics, (), 281-300 · Zbl 0652.76021 [5] Saad, Y.; Schultz, M.H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. scient. stat. comput., 7, 856-869, (1986) · Zbl 0599.65018 [6] Tezduyar, T.E.; Liou, J., Grouped element-by-element iteration schemes for incompressible flow computations, Comput. phys. comm., 53, 441-453, (1989) · Zbl 0803.76065 [7] Tezduyar, T.E.; Liou, J.; Nguyen, T.; Poole, S., Adaptive implicit-explicit and parallel element-by-element factorization schemes, (), 443-463 · Zbl 0674.76079 [8] Shakib, F.; Hughes, T.J.R.; Johan, Z., A multi-element group preconditioned GMRES algorithm for nonsymmetric systems arising in finite element analysis, Comput. methods appl. mech. engrg., 75, 415-456, (1989) · Zbl 0687.76065 [9] Liou, J.; Tezduyar, T.E., Iterative adaptive implicit-explicit methods for flow problems, Internat. J. numer. methods fluids, 11, 867-880, (1990) · Zbl 0704.76031 [10] Liou, J.; Tezduyar, T.E., A clustered element-by-element iteration method for finite element computations, University of minnesota supercomputer institute research report UMSI 91/279, (1991) · Zbl 0766.65090 [11] Liou, J.; Tezduyar, T.E., Computation of compressible and incompressible flows with the clustered element-by-element method, University of minnesota supercomputer institute research report UMSI 90/215, (1990) · Zbl 0716.76049 [12] Tezduyar, T.E.; Mittal, S., Finite element computation of incompressible flows, University of minnesota supercomputer institute research report UMSI 91/152, (1991) · Zbl 0846.76048 [13] Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, (), 140-150, Chapter 13 [14] Meurant, G., A domain decomposition method for parabolic problems, (1989), preprint
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.