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.


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