  Domain decomposition (DD) methods are widely used as preconditioner
techniques. Their effectiveness relies on the choice of a locally constructed
coarse space. Thus far, this construction was mostly achieved using
non-assembled matrices from discretized partial differential equations (PDEs).
Therefore, DD methods were mainly successful when solving systems stemming from
PDEs. In this paper, we present a fully algebraic multilevel DD method where
the coarse space can be constructed locally and efficiently without any
information besides the coefficient matrix. The condition number of the
preconditioned matrix can be bounded by a user-prescribed number. Numerical
experiments illustrate the effectiveness of the preconditioner on a range of
problems arising from different applications.
