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BDDC algorithms for incompressible Stokes equations. (English) Zbl 1233.76077

Summary: The purpose of this paper is to extend the balancing domain decomposition by constraints (BDDC) algorithm to saddle-point problems that arise when mixed finite element methods are used to approximate the system of incompressible Stokes equations. The BDDC algorithms are iterative substructuring methods which form a class of domain decomposition methods based on the decomposition of the domain of the differential equations into nonoverlapping subdomains. They are defined in terms of a set of primal continuity constraints which are enforced across the interface between the subdomains and which provide a coarse space component of the preconditioner. Sets of such constraints are identified for which bounds on the rate of convergence can be established that are just as strong as previously known bounds for the elliptic case. In fact, the preconditioned operator is effectively positive definite, which makes the use of a conjugate gradient method possible. A close connection is also established between the BDDC and dual-primal finite element tearing and interconnecting (FETI-DP) algorithms for the Stokes case.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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