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By way of introduction,a short survey of preconditioning methods for the
iterative solution of large scale linear systems of equations is given.
The most efficient iteration methods are based on some form of a
(generalied) conjugate gradient method.The purpose of preconditioning is
then to significantly increase its rate of convergence.This can be done
by choosing preconditionings which result in a favourable distribution
of eigenvalues of the preconditioned matrix.
Early versions of preconditioning methods used incomplete factorization methods,in this way bridging the gap between unpreconditioned iterative and direct solution methods. Methods based on block partitionings of the given matrix are in general most efficient and recursive use of a certain two-by-two block partitioning has been shown to offer (nearly) optimal order convergence rates,i.e., a number of iterations which depend at most logarithmically on the size of the problem, when applied for finite element matrices. The partitioning is normally based on splitting the node set in coarse mesh points and (remaining) fine mesh mesh points and ,by recursion,the coarse mesh set is similarly partitioned in smaller sets. Both classical smoothing and coarse grid correction multi-grid methods and purely algebraic multilevel methods can be formulated in this framework. The talk will deal mainly with a special variant of a preconditioning method for matrices on two by two block form, which method can be particularly advantageous for non-symmetric matrices. Such methods require approximations of both the pivot block(fine grid) matrix and the Schur complement(coarse grid) matrix. The efficiency in using element-by-element techniques to construct the preconditioners for both of these matrices is emphasized. |