We focus on the Schwinger model which is an ideal laboratory to develop and to test preconditioners for discretizations of lattice fermions since it has many features in common with lattice QCD. The corresponding Schwinger matrices represent a nearest neighbor coupling on a regular two-dimensional lattice. In general, the Schwinger matrices are large, sparse and structured but totally disordered.
In contrast to the original system, an initial odd-even reduction enables us to distinguish between strong and weak edges in the digraph induced by the odd-even reduced system. The F-sites are chosen to be a maximal independent vertex set of the reduced digraph consisting of the vertex set and the set of strong edges. Although the F-sites are selected algebraically, the resulting F/C partitioning of the even lattice sites can be easily fixed in advance such that costly coarsening strategies become obsolete.
To precondition the subsystem residing on the F-sites only, standard incomplete LU preconditioners (ILU) are considered. Rigorous bounds of the condition number of the preconditioned submatrix are given. An appropriate and easy to construct Schur complement approximation is proposed. With respect to the odd-even reduced system no fill-in occurs so that the Schur complement preconditioner allows a recursive procedure on coarse lattices.
The algebraic multilevel preconditioner is compared with standard odd-even preconditioning. Numerical experiments indicate that the Schur complement preconditioner outperforms the standard odd-even preconditioner.