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

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.

[1]  T. Manteuffel.  The Tchebychev iteration for nonsymmetric linear systems , 1977 .

[2]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[3]  G. Golub,et al.  The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems , 1988 .

[4]  H. Walker,et al.  GMRES On (Nearly) Singular Systems , 1997, SIAM J. Matrix Anal. Appl..

[5]  L. Pavarino,et al.  Overlapping Schwarz methods for mixed linear elasticity and Stokes problems , 1998 .

[6]  Charbel Farhat,et al.  A scalable dual-primal domain decomposition method , 2000, Numerical Linear Algebra with Applications.

[7]  Luca F. Pavarino,et al.  A comparison of overlapping Schwarz methods and block preconditioners for saddle point problems , 2000, Numer. Linear Algebra Appl..

[8]  D. Rixen,et al.  FETI‐DP: a dual–primal unified FETI method—part I: A faster alternative to the two‐level FETI method , 2001 .

[9]  O. Widlund,et al.  Balancing Neumann‐Neumann methods for incompressible Stokes equations , 2001 .

[10]  Olof B. Widlund,et al.  DUAL-PRIMAL FETI METHODS FOR THREE-DIMENSIONAL ELLIPTIC PROBLEMS WITH HETEROGENEOUS COEFFICIENTS , 2022 .

[11]  Jiao Li.  Dual-primal FETI methods for stationary Stokes and Navier--Stokes equations , 2002 .

[12]  CLARK R. DOHRMANN,et al.  A Preconditioner for Substructuring Based on Constrained Energy Minimization , 2003, SIAM J. Sci. Comput..

[13]  Clark R. Dohrmann,et al.  Convergence of a balancing domain decomposition by constraints and energy minimization , 2002, Numer. Linear Algebra Appl..

[14]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[15]  Chang-Ock Lee,et al.  A Preconditioner for the FETI-DP Formulation with Mortar Methods in Two Dimensions , 2004, SIAM J. Numer. Anal..

[16]  Jing Li,et al.  A Dual-Primal FETI method for incompressible Stokes equations , 2005, Numerische Mathematik.

[17]  J. Mandel,et al.  An algebraic theory for primal and dual substructuring methods by constraints , 2005 .

[18]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[20]  Olof B. Widlund,et al.  Dual‐primal FETI methods for linear elasticity , 2006 .

[21]  Olof B. Widlund,et al.  FETI‐DP, BDDC, and block Cholesky methods , 2006 .

[22]  Chang-Ock Lee,et al.  A Neumann-Dirichlet Preconditioner for a FETI-DP Formulation of the Two-Dimensional Stokes Problem with Mortar Methods , 2006, SIAM J. Sci. Comput..

[23]  Olof B. Widlund,et al.  BDDC Algorithms for Incompressible Stokes Equations , 2006, SIAM J. Numer. Anal..

[24]  Axel Klawonn,et al.  Inexact FETI‐DP methods , 2007 .

[25]  Clark R. Dohrmann,et al.  An approximate BDDC preconditioner , 2007, Numer. Linear Algebra Appl..

[26]  Susanne C. Brenner,et al.  BDDC and FETI-DP without matrices or vectors , 2007 .

[27]  O. Widlund,et al.  On the use of inexact subdomain solvers for BDDC algorithms , 2007 .

[28]  Olof B. Widlund,et al.  A BDDC Method for Mortar Discretizations Using a Transformation of Basis , 2008, SIAM J. Numer. Anal..

[29]  Hyea Hyun Kim,et al.  A FETI-DP FORMULATION OF THREE DIMENSIONAL ELASTICITY PROBLEMS WITH MORTAR DISCRETIZATION , 2005 .

[30]  Olof B. Widlund,et al.  An Analysis of a FETI-DP Algorithm on Irregular Subdomains in the Plane , 2008, SIAM J. Numer. Anal..

[31]  Olof B. Widlund,et al.  An Overlapping Schwarz Algorithm for Almost Incompressible Elasticity , 2009, SIAM J. Numer. Anal..

[32]  O. Widlund,et al.  Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity , 2009 .

[33]  Chang-Ock Lee,et al.  A FETI-DP Formulation for the Stokes Problem without Primal Pressure Components , 2010, SIAM J. Numer. Anal..

[34]  Chang-Ock Lee,et al.  On the selection of primal unknowns for a FETI-DP formulation of the Stokes problem in two dimensions , 2010, Comput. Math. Appl..

[35]  Luca F. Pavarino,et al.  Robust BDDC Preconditioners for Reissner-Mindlin Plate Bending Problems and MITC Elements , 2010, SIAM J. Numer. Anal..

[36]  Olof B. Widlund,et al.  BDDC Preconditioners for Spectral Element Discretizations of Almost Incompressible Elasticity in Three Dimensions , 2010, SIAM J. Sci. Comput..

[37]  Chang-Ock Lee,et al.  A FETI--DP Formulation for the Three-Dimensional Stokes Problem without Primal Pressure Unknowns , 2010, SIAM J. Sci. Comput..

[38]  Olof B. Widlund,et al.  BDDC and FETI-DP Preconditioners for Spectral Element Discretizations of Almost Incompressible Elasticity , 2011 .