A three-level BDDC algorithm for a saddle point problem

BDDC algorithms have previously been extended to the saddle point problems arising from mixed formulations of elliptic and incompressible Stokes problems. In these two-level BDDC algorithms, all iterates are required to be in a benign space, a subspace in which the preconditioned operators are positive definite. This requirement can lead to large coarse problems, which have to be generated and factored by a direct solver at the beginning of the computation and they can ultimately become a bottleneck. An additional level is introduced in this paper to solve the coarse problem approximately and to remove this difficulty. This three-level BDDC algorithm keeps all iterates in the benign space and the conjugate gradient methods can therefore be used to accelerate the convergence. This work is an extension of the three-level BDDC methods for standard finite element discretization of elliptic problems and the same rate of convergence is obtained for the mixed formulation of the same problems. Estimate of the condition number for this three-level BDDC methods is provided and numerical experiments are discussed.

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

[2]  Clark R. Dohrmann,et al.  On Multilevel BDDC , 2008 .

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

[4]  Eric S. Lander,et al.  AN ALGEBRAIC APPROACH , 1983 .

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

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

[7]  Susanne C. Brenner,et al.  Lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions , 2000, Math. Comput..

[8]  Barbara I. Wohlmuth,et al.  An Iterative Substructuring Method for Raviart-Thomas Vector Fields in Three Dimensions , 2000, SIAM J. Numer. Anal..

[9]  Clark R. Dohrmann,et al.  Multispace and multilevel BDDC , 2007, Computing.

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

[11]  Olof B. Widlund,et al.  Bddc domain decomposition algorithms: methods with three levels and for flow in porous media , 2006 .

[12]  Non-overlapping Domain Decomposition Preconditioners with Inexact Solves , 1998 .

[13]  Barry Smith.  A Parallel Implementation of an Iterative Substructuring Algorithm for Problems in Three Dimensions , 1993, SIAM J. Sci. Comput..

[14]  Xiao-Chuan Cai,et al.  Parallel fully coupled Schwarz preconditioners for saddle point problems. , 2006 .

[15]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[16]  Olof B. Widlund,et al.  A Domain Decomposition Method with Lagrange Multipliers and Inexact Solvers for Linear Elasticity , 2000, SIAM J. Sci. Comput..

[17]  Arnd Meyer,et al.  The approximate Dirichlet Domain Decomposition method. Part I: An algebraic approach , 1991, Computing.

[18]  Barbara I. Wohlmuth,et al.  Discretization Methods and Iterative Solvers Based on Domain Decomposition , 2001, Lecture Notes in Computational Science and Engineering.

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

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

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

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

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

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

[25]  Christoph Börgers,et al.  The Neumann-Dirichlet domain decomposition method with inexact solvers on the subdomains , 1989 .

[26]  A. Toselli.  Domain Decomposition Methods for Vector Field Problems , 1999 .

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

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

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

[30]  Arnd Meyer,et al.  The approximate Dirichlet Domain Decomposition method. Part II: Applications to 2nd-order Elliptic B.V.P.s , 2005, Computing.

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

[32]  O. Widlund,et al.  Domain Decomposition Methods in Science and Engineering XVI , 2007 .

[33]  Barbara I. Wohlmuth,et al.  An iterative substructuring method for Maxwell's equations in two dimensions , 2001, Math. Comput..

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

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

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

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