A two-level nonoverlapping Schwarz algorithm for the Stokes problem without primal pressure unknowns

• Published in: International journal for numerical methods in engineering Vol. 88; no. 13; pp. 1390 - 1410
• Main Authors: Kim, Hyea Hyun, Lee, Chang-Ock
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 30.12.2011; Wiley
• Subjects: Algorithms; BDDC; Coarsening; Computational methods in fluid dynamics; Computational techniques; Corners; Eigenvalues; Exact sciences and technology; FETI-DP; Fluid dynamics; Fluid flow; Fundamental areas of phenomenology (including applications); Mathematical methods in physics; Mathematical models; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis; Numerical analysis. Scientific computation; Physics; preconditioner; Sciences and techniques of general use; Stokes problem; Three dimensional; two-level Schwarz algorithm; Stresses; Balancing; Stokes problem; two-level Schwarz algorithm; Wedge; Finite element method; Incompressible flow; FETI-DP; Primal dual method; Problem solving; Vibration control; Modelling; Mixed problem; Eigenvalue problem; preconditioner; Domain decomposition; Preconditioning; Numerical convergence; BDDC
• ISSN: 0029-5981; 1097-0207; 1097-0207
• DOI: 10.1002/nme.3227

A two‐level nonoverlapping Schwarz algorithm is developed for the Stokes problem. The main feature of the algorithm is that a mixed problem with both velocity and pressure unknowns is solved with a balancing domain decomposition by constraints (BDDC)‐type preconditioner, which consists of solving local Stokes problems and one global coarse problem related to only primal velocity unknowns. Our preconditioner allows to use a smaller set of primal velocity unknowns than other BDDC preconditioners without much concern on certain flux conditions on the subdomain boundaries and the inf–sup stability of the coarse problem. In the two‐dimensional case, velocity unknowns at subdomain corners are selected as the primal unknowns. In addition to them, averages of each velocity component across common faces are employed as the primal unknowns for the three‐dimensional case. By using its close connection to the Dual–primal finite element tearing and interconnecting (FETI‐DP algorithm) (SIAM J Sci Comput 2010; 32: 3301–3322; SIAM J Numer Anal 2010; 47: 4142–4162], it is shown that the resulting matrix of our algorithm has the same eigenvalues as the FETI‐DP algorithm except zero and one. The maximum eigenvalue is determined by H/h, the number of elements across each subdomains, and the minimum eigenvalue is bounded below by a constant, which does not depend on any mesh parameters. Convergence of the method is analyzed and numerical results are included. Copyright © 2011 John Wiley & Sons, Ltd.