ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2012, Vol. 52, No. 9, pp. 1260–1294. © Pleiades Publishing, Ltd., 2012. 1260 INTRODUCTION In the famous series of papers on the method of domain decomposition, Bramble, Pasciak & Schatz (see [3–6]) presented an efficient and rather general DD (domain decomposition) preconditioner for h finite element discretizations of 3D elliptic partial differential equations. Since then, it is commonly referred as BPS preconditioner with the abbreviation coming from first letters in surnames of the authors. BPS preconditioner has been the origin of the whole family of DD (domain decomposition) Dirichlet- Dirichlet type efficient preconditioners-solvers as for h so hp discretizations of elliptic problems. At first, it was expanded to more general h discretizations and domain decompositions with the state of art in this area well reflected in recent books by Smith. Biorstad & Gropp (see [39]) and Toselli & Widlund (see [40]) see also the review paper by Korneev & Langer (see [20]). Vast bibliography of related papers can be also found in these publications. For the developments of fast DD preconditioners-solvers of BPS type for hp discretizations of 3D elliptic problems, apart from the cited books we refer to Pavarmo & Widlund (see [34–36]), Casarin (see [10], Korneev, Langer & Xanthis (see [21, 22]), Korneev & Rytov (see [24–26]). As we noted, generalizations of BPS preconditioners were related not only to the types of finite element discretizations, but also to domain decompositions. Bramble et al. assumed that subdomains are images On Domain Decomposition Preconditioner of BPS Type for Finite Element Discretizations of 3D Elliptic Equations 1 V. G. Korneev St. Petersburg State University, Universitetskaya nab. 7–9, St. Petersburg, 199034 Russia St. Petersburg State Polytechnical University, Polytechnicheskaya ul. 29, St. Petersburg, 195251 Russia e-mail: VadimKorneev@yahoo.com Received January 14, 2011; in final form, October 29, 2011 Abstract—BPS is a well known an efficient and rather general domain decomposition Dirichlet- Dirichlet type preconditioner, suggested in the famous series of papers Bramble, Pasciak and Schatz (1986–1989). Since then, it has been serving as the origin for the whole family of domain decomposi- tion Dirichlet-Dirichlet type preconditioners-solvers as for h so hp discretizations of elliptic problems. For its original version, designed for h discretizations, the named authors proved the bound (1 + log 2 H/h) for the relative condition number under some restricting conditions on the domain decomposition and finite element discretization. Here H/h is the maximal relation of the characteristic size H of a decomposition subdomain to the mesh parameter h of its discretization. It was assumed that subdomains are images of the reference unite cube by trilinear mappings. Later sim- ilar bounds related to h discretizations were proved for more general domain decompositions, defined by means of coarse tetrahedral meshes. These results, accompanied by the development of some spe- cial tools of analysis aimed at such type of decompositions, were summarized in the book of Toselli and Widlund (2005). This paper is also confined to h discretizations. We further expand the range of admis- sible domain decompositions for constructing BPS preconditioners, in which decomposition subdo- mains can be convex polyhedrons, satisfying some conditions of shape regularity. We prove the bound for the relative condition number with the same dependence on H/h as in the bound given above. Along the way to this result, we simplify the proof of the so called abstract bound for the relative con- dition number of the domain decomposition preconditioner. In the part, related to the analysis of the interface sub-problem preconditioning, our technical tools are generalization of those used by Bram- ble, Pasciak and Schatz. 2000 Mathematics Subject Classification: 65N22; 65M30; 65N55. DOI: 10.1134/S0965542512090059 Keywords: domain decomposition method, preconditioning, fast solvers, finite element methods, 3D elliptic equations. 1 The author has been partially supported by the Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences during his research visits at Linz. The article is published in the original.