Geodesic
Renumbering strategies based on multi-level techniques combined with ILU-decompositions
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 37 (1997) no. 11, pp. 1294-1300

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In this paper we present an incomplete factorization technique which uses a renumbering of the unknowns, based on a sequence of grids as in multi-grid. For many problems discretised on structured grids, we obtain almost grid-independent convergence when this factorization is combined with some conjugate gradient-like method. Also, a similar preconditioning technique is described which can be used for matrices with arbitrary sparsity patterns as those arising from finite element methods on unstructured grids. During the factorization we use a reordering to guarantee that the diagonal blocks to be inverted remain strongly diagonally dominant. This makes it possible to approximate the needed inverses by only a diagonal matrix, leading to more potential parallelism. The method is demonstrated for a number of test problems and compared to some standard methods. 
@article{ZVMMF_1997_37_11_a1,
     author = {E. F. F. Botta and A. van der Ploeg},
     title = {Renumbering strategies based on multi-level techniques combined with {ILU-decompositions}},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1294--1300},
     publisher = {mathdoc},
     volume = {37},
     number = {11},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_1997_37_11_a1/}
}
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E. F. F. Botta; A. van der Ploeg. Renumbering strategies based on multi-level techniques combined with ILU-decompositions. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 37 (1997) no. 11, pp. 1294-1300. http://geodesic.mathdoc.fr/item/ZVMMF_1997_37_11_a1/