Toward a new fully algebraic preconditioner for symmetric positive definite problems

Abstract: A new domain decomposition preconditioner is introduced for efficiently solving linear systems Ax = b with a symmetric positive definite matrix A. The particularity of the new preconditioner is that it is not necessary to have access to the so-called Neumann matrices (i.e.: the matrices that result from assembling the variational problem underlying A restricted to each subdomain). All the components in the preconditioner can be computed with the knowledge only of A (and this is the meaning given here to the word algebraic). The new preconditioner relies on the GenEO coarse space for a matrix that is a low-rank modification of A and on the Woodbury matrix identity. The idea underlying the new preconditioner is introduced here for the first time with a first version of the preconditioner. Some numerical illustrations are presented. A more extensive presentation including some improved variants of the new preconditioner can be found in 7.