A non-overlapping domain decomposition method for incompressible Stokes equations with continuous pressure

A non-overlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible Stokes equations. A continuous finite element space for the pressure is used. In the proposed algorithm, Lagrange multipliers are used to enforce continuity of the velocity component across the subdomain domain boundary. The continuity of the pressure component is enforced in the primal form, i.e., neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no Lagrange multipliers are needed. After eliminating all velocity variables and the independent subdomain interior parts of the pressures, a symmetric positive semi-definite linear system for the subdomain boundary pressures and the Lagrange multipliers is formed and solved by a preconditioned conjugate gradient method. A lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments demonstrate the convergence rate of the proposed algorithm.

[1]  P. Hood,et al.  A numerical solution of the Navier-Stokes equations using the finite element technique , 1973 .

[2]  C. Farhat,et al.  A method of finite element tearing and interconnecting and its parallel solution algorithm , 1991 .

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

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

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

[6]  P. Goldfeld.  Balancing Neumann-Neumann Preconditioners for the Mixed Formulation of Almost-Incompressible Linear Elasticity , 2003 .

[7]  Olof B. Widlund,et al.  Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity , 2003, Numerische Mathematik.

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

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

[10]  X. Tu.  A BDDC ALGORITHM FOR A MIXED FORMULATION OF FLOW IN POROUS MEDIA , 2005 .

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

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

[13]  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..

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

[15]  C. Dohrmann.  Preconditioning of Saddle Point Systems by Substructuring and a Penalty Approach , 2007 .

[16]  X. Tu.  A BDDC algorithm for flow in porous media with a hybrid finite element discretization. , 2007 .

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

[18]  Gary R. Consolazio,et al.  Finite Elements , 2007, Handbook of Dynamic System Modeling.

[19]  Y. Maday,et al.  Optimal convergence properties of the FETI domain decomposition method , 2007 .

[20]  Xuemin Tu.  Three-level BDDC , 2007 .

[21]  Xuemin Tu,et al.  Three‐level BDDC in two dimensions , 2007 .

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

[23]  Xuemin Tu.  Three-Level BDDC in Three Dimensions , 2007, SIAM J. Sci. Comput..

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

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

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

[27]  Xuemin Tu,et al.  A Three-Level BDDC Algorithm for Mortar Discretizations , 2009, SIAM J. Numer. Anal..

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

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

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

[31]  Xuemin Tu,et al.  A three-level BDDC algorithm for a saddle point problem , 2008, Numerische Mathematik.

[32]  Hyea-Hyun Kim,et al.  A two‐level nonoverlapping Schwarz algorithm for the Stokes problem without primal pressure unknowns , 2011 .

[33]  J. Mandel,et al.  Application of the parallel BDDC preconditioner to the Stokes flow , 2011, 1101.1775.

[34]  Hani Benhassine,et al.  A non-overlapping domain decomposition method for continuous-pressure mixed finite element approximations of the Stokes problem , 2011 .