A FETI-DP Formulation for the Stokes Problem without Primal Pressure Components

A scalable FETI-DP (dual-primal finite element tearing and interconnecting) algorithm for the Stokes problem that employs a lumped preconditioner is developed and analyzed. A pair of inf-sup stable velocity and pressure finite element spaces is used to obtain a discrete problem. Differently from previous approaches, no primal pressure unknowns are selected and only velocity primal unknowns at subdomain corners are selected. This leads to a symmetric and positive definite coarse problem matrix in the FETI-DP operator, while a larger and indefinite coarse problem appears in the previous approaches. In addition, its condition number bound is proved to be the same as the FETI-DP algorithm with a lumped preconditioner for elliptic problems. Numerical results are included.

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

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

[3]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

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

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

[6]  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..

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

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

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

[10]  C. Farhat,et al.  Optimal convergence properties of the FETI domain decomposition method , 1994 .

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

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

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

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

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

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

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

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

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