On domain decomposition preconditioner of BPS type for finite element discretizations of 3D elliptic equations

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 decomposition 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 $O(1+\log^2H/h)$ 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 similar 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 special 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 admissible domain decompositions for constructing BPS preconditioners, in which decomposition subdomains 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 condition 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 Bramble, Pasciak and Schatz.