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Balancing Neumann‐Neumann methods for incompressible Stokes equations
Author(s) -
Pavarino Luca F.,
Widlund Olof B.
Publication year - 2002
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10020
Subject(s) - mathematics , preconditioner , neumann boundary condition , mathematical analysis , discretization , bounded function , schur complement , domain decomposition methods , navier–stokes equations , domain (mathematical analysis) , boundary (topology) , finite element method , compressibility , linear system , eigenvalues and eigenvectors , physics , thermodynamics , aerospace engineering , engineering , quantum mechanics
Balancing Neumann‐Neumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the interior velocity component and all except the constant‐pressure component, of each subdomain problem are implicitly eliminated. The resulting saddle point Schur complement is solved with a Krylov space method with a balancing Neumann‐Neumann preconditioner based on the solution of a coarse Stokes problem with a few degrees of freedom per subdomain and on the solution of local Stokes problems with natural and essential boundary conditions on the subdomains. This preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively. The condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and a factor that depends on the inverse of the inf‐sup constants of the discrete problem and of the coarse subproblem. Numerical results show that the method is quite fast; they are also fully consistent with the theory. © 2002 John Wiley & Sons, Inc.