A unified dual-primal finite element tearing and interconnecting approach for incompressible Stokes equations

SUMMARYA unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it...

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Veröffentlicht in:	International journal for numerical methods in engineering Jg. 94; H. 2; S. 128 - 149
Hauptverfasser:	Tu, Xuemin, Li, Jing
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Chichester Blackwell Publishing Ltd 13.04.2013 Wiley Wiley Subscription Services, Inc
Schlagworte:	Algorithms Approximation BDDC Computational methods in fluid dynamics Convergence Dirichlet problem domain decomposition Exact sciences and technology FETI-DP Finite element method Fluid dynamics Fluid flow Fundamental areas of phenomenology (including applications) incompressible Stokes Mathematical analysis Mathematics Methods of scientific computing (including symbolic computation, algebraic computation) Numerical analysis. Scientific computation Physics Sciences and techniques of general use Tearing Performance evaluation Linear systems Stokes equation Conjugate gradient methods Experimental study Mixed method Finite element method Incompressible flow FETI-DP Primal dual method Convergence rate Dirichlet problem Boundary-value problems Modelling Incompressible fluid Domain decomposition Preconditioning incompressible Stokes BDDC
ISSN:	0029-5981, 1097-0207
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Abstract	SUMMARYA unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it allows using both continuous and discontinuous pressures in the algorithm, whereas previous FETI‐DP methods only apply to discontinuous pressures. A preconditioned conjugate gradient method is used in the algorithm with either a lumped or a Dirichlet preconditioner, and scalable convergence rates are proved. This framework is also used to describe several previously developed FETI‐DP algorithms and greatly simplifies their analysis. Numerical experiments of solving a two‐dimensional incompressible Stokes problem demonstrate the performances of the discussed FETI‐DP algorithms represented under the same framework.Copyright © 2012 John Wiley & Sons, Ltd.
AbstractList	SUMMARY A unified framework of dual-primal finite element tearing and interconnecting (FETI-DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it allows using both continuous and discontinuous pressures in the algorithm, whereas previous FETI-DP methods only apply to discontinuous pressures. A preconditioned conjugate gradient method is used in the algorithm with either a lumped or a Dirichlet preconditioner, and scalable convergence rates are proved. This framework is also used to describe several previously developed FETI-DP algorithms and greatly simplifies their analysis. Numerical experiments of solving a two-dimensional incompressible Stokes problem demonstrate the performances of the discussed FETI-DP algorithms represented under the same framework.Copyright © 2012 John Wiley & Sons, Ltd. [PUBLICATION ABSTRACT] A unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it allows using both continuous and discontinuous pressures in the algorithm, whereas previous FETI‐DP methods only apply to discontinuous pressures. A preconditioned conjugate gradient method is used in the algorithm with either a lumped or a Dirichlet preconditioner, and scalable convergence rates are proved. This framework is also used to describe several previously developed FETI‐DP algorithms and greatly simplifies their analysis. Numerical experiments of solving a two‐dimensional incompressible Stokes problem demonstrate the performances of the discussed FETI‐DP algorithms represented under the same framework.Copyright © 2012 John Wiley & Sons, Ltd. SUMMARY A unified framework of dual-primal finite element tearing and interconnecting (FETI-DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it allows using both continuous and discontinuous pressures in the algorithm, whereas previous FETI-DP methods only apply to discontinuous pressures. A preconditioned conjugate gradient method is used in the algorithm with either a lumped or a Dirichlet preconditioner, and scalable convergence rates are proved. This framework is also used to describe several previously developed FETI-DP algorithms and greatly simplifies their analysis. Numerical experiments of solving a two-dimensional incompressible Stokes problem demonstrate the performances of the discussed FETI-DP algorithms represented under the same framework.Copyright [copy 2012 John Wiley & Sons, Ltd. SUMMARYA unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it allows using both continuous and discontinuous pressures in the algorithm, whereas previous FETI‐DP methods only apply to discontinuous pressures. A preconditioned conjugate gradient method is used in the algorithm with either a lumped or a Dirichlet preconditioner, and scalable convergence rates are proved. This framework is also used to describe several previously developed FETI‐DP algorithms and greatly simplifies their analysis. Numerical experiments of solving a two‐dimensional incompressible Stokes problem demonstrate the performances of the discussed FETI‐DP algorithms represented under the same framework.Copyright © 2012 John Wiley & Sons, Ltd.
Author	Tu, Xuemin Li, Jing
Author_xml	– sequence: 1 givenname: Xuemin surname: Tu fullname: Tu, Xuemin email: Correspondence to: Xuemin Tu, Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045-7594, USA., xtu@math.ku.edu organization: Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, KS 66045-7594, Lawrence, USA – sequence: 2 givenname: Jing surname: Li fullname: Li, Jing organization: Department of Mathematical Sciences, Kent State University, OH 44242, Kent, USA
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Keywords	Performance evaluation Linear systems Stokes equation Conjugate gradient methods Experimental study Mixed method Finite element method Incompressible flow FETI-DP Primal dual method Convergence rate Dirichlet problem Boundary-value problems Modelling Incompressible fluid Domain decomposition Preconditioning incompressible Stokes BDDC
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References	Bramble JH, Pasciak JE. A domain decomposition technique for Stokes problems. Applied Numerical Mathematics 1989/90; 6:251-261. Tu X. Three-level BDDC. Lecture Notes in Computational Science and Engineering 2007; 55:437-444. Kim HH, Lee CO, Park EH. A FETI-DP formulation for the Stokes problem without primal pressure components. SIAM Journal on Numerical Analysis 2010; 47:4142-4162. Goldfeld P, Pavarino LF, Widlund OB. Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity. Numerische Mathematik 2003; 95:283-324. Dohrmann CR, Widlund OB. Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity. International Journal for Numerical Methods in Engineering 2010; 82:157-183. Tu X. BDDC algorithm for a mixed formulation of flows in porous media. Electronic Transactions on Numerical Analysis 2005; 20:164-179. Toselli A, Widlund OB. Domain Decomposition Methods - Algorithms and Theory. Springer-Verlag: New York, 2004. Farhat C. A Lagrange multiplier based divide and conquer finite element algorithm. Computing Systems in Engineering 1991; 2:149-156. Farhat C, Lesoinne M, Pierson K. A scalable dual-primal domain decomposition method. Numerical Linear Algebra with Applications 2000; 7:687-714. Li J. A dual-primal FETI method for incompressible Stokes equations. Numerische Mathematik 2005; 102:257-275. Tu X. Three-level BDDC in three dimensions. SIAM Journal on Scientific Computing 2007; 29:1759-1780. Tu X. Three-level BDDC in two dimensions. International Journal for Numerical Methods in Engineering 2007; 69:33-59. Dohrmann CR. Preconditioning of saddle point systems by substructuring and a penalty approach. In Lecture Notes in Computational Science and Engineering 2006; 55:53-64. Li J, Widlund OB. BDDC algorithms for incompressible Stokes equations. SIAM Journal on Numerical Analysis 2006; 44:2432-2455. Kim HH, Lee CO. A Neumann-Dirichlet preconditioner for a FETI-DP formulation of the two-dimensional Stokes problem with mortar methods. SIAM Journal on Scientific Computing 2006; 28:1133-1152. Tu X. A BDDC algorithm for flow in porous media with a hybrid finite element discretization. Electronic Transactions on Numerical Analysis 2007; 26:146-160. Farhat C, Roux FX. An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems. SIAM Journal on Scientific and Statistical Computing 1992; 13:379-396. Li J, Widlund OB. FETI-DP, BDDC, and block Cholesky methods. International Journal for Numerical Methods in Engineering 2006; 66:250-271. Li J, Widlund OB. On the use of inexact subdomain solvers for BDDC algorithms. Computer Methods in Applied Mechanics and Engineering 2007; 196:1415-1428. Pavarino LF, Widlund OB, Zampini S. BDDC preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions. SIAM Journal on Scientific Computing 2010; 32:3604-3626. Mandel J, Tezaur R. On the convergence of a dual-primal substructuring method. Numerische Mathematik 2001; 88:543-558. Farhat C, Mandel J, Roux FX. Optimal convergence properties of the FETI domain decomposition method. Computer Methods in Applied Mechanics and Engineering 1994; 115:367-388. Klawonn A, Widlund OB, Dryja M. Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients. SIAM Journal on Numerical Analysis 2002; 40:159-179. Klawonn A, Pavarino LF. Overlapping Schwarz methods for mixed linear elasticity and Stokes problems. Computer Methods in Applied Mechanics and Engineering 1998; 165:233-245. Farhat C, Roux FX. A method of finite element tearing and interconnecting and its parallel solution algorithm. International Journal for Numerical Methods in Engineering 1991; 32:1205-1227. Farhat C, Lesoinne M, Le Tallec P, Pierson K, Rixen D. FETI-DP: a dual-primal unified FETI method - part I: a faster alternative to the two-level FETI method. International Journal for Numerical Methods in Engineering 2001; 50:1523-1544. Dohrmann CR. An approximate BDDC preconditioner. Numerical Linear Algebra with Applications 2007; 14:149-168. Tu X. A three-level BDDC algorithm for saddle point problems. Numerische Mathematik 2011; 119:189-217. Dohrmann CR, Widlund OB. An overlapping Schwarz algorithm for almost incompressible elasticity. SIAM Journal on Numerical Analysis 2009; 47:2897-2923. Braess D. Finite Elements, 2nd ed. Cambridge: Cambridge, UK, 2001. Klawonn A, Rheinbach O. Inexact FETI-DP methods. International Journal for Numerical Methods in Engineering 2007; 69:284-307. Pavarino LF, Widlund OB. Balancing Neumann-Neumann methods for incompressible Stokes equations. Communications on Pure and Applied Mathematics 2002; 55:302-335. Kim HH, Tu X. A three-level BDDC algorithm for mortar discretization. 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References_xml	– reference: Toselli A, Widlund OB. Domain Decomposition Methods - Algorithms and Theory. Springer-Verlag: New York, 2004. – reference: Dohrmann CR, Widlund OB. Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity. International Journal for Numerical Methods in Engineering 2010; 82:157-183. – reference: Goldfeld P, Pavarino LF, Widlund OB. Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity. Numerische Mathematik 2003; 95:283-324. – reference: Klawonn A, Widlund OB, Dryja M. Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients. SIAM Journal on Numerical Analysis 2002; 40:159-179. – reference: Tu X. Three-level BDDC in three dimensions. SIAM Journal on Scientific Computing 2007; 29:1759-1780. – reference: Klawonn A, Widlund OB. Dual-primal FETI methods for linear elasticity. Communications on Pure and Applied Mathematics 2006; 59:1523-1572. – reference: Dohrmann CR. An approximate BDDC preconditioner. Numerical Linear Algebra with Applications 2007; 14:149-168. – reference: Farhat C, Mandel J, Roux FX. Optimal convergence properties of the FETI domain decomposition method. Computer Methods in Applied Mechanics and Engineering 1994; 115:367-388. – reference: Mandel J, Tezaur R. On the convergence of a dual-primal substructuring method. Numerische Mathematik 2001; 88:543-558. – reference: Farhat C, Lesoinne M, Le Tallec P, Pierson K, Rixen D. FETI-DP: a dual-primal unified FETI method - part I: a faster alternative to the two-level FETI method. International Journal for Numerical Methods in Engineering 2001; 50:1523-1544. – reference: Klawonn A, Pavarino LF. Overlapping Schwarz methods for mixed linear elasticity and Stokes problems. Computer Methods in Applied Mechanics and Engineering 1998; 165:233-245. – reference: Pavarino LF, Widlund OB. Balancing Neumann-Neumann methods for incompressible Stokes equations. Communications on Pure and Applied Mathematics 2002; 55:302-335. – reference: Tu X. Three-level BDDC. Lecture Notes in Computational Science and Engineering 2007; 55:437-444. – reference: Tu X. A three-level BDDC algorithm for saddle point problems. Numerische Mathematik 2011; 119:189-217. – reference: Klawonn A, Rheinbach O. Inexact FETI-DP methods. International Journal for Numerical Methods in Engineering 2007; 69:284-307. – reference: Dohrmann CR. Preconditioning of saddle point systems by substructuring and a penalty approach. In Lecture Notes in Computational Science and Engineering 2006; 55:53-64. – reference: Tu X. A BDDC algorithm for flow in porous media with a hybrid finite element discretization. Electronic Transactions on Numerical Analysis 2007; 26:146-160. – reference: Li J, Widlund OB. FETI-DP, BDDC, and block Cholesky methods. International Journal for Numerical Methods in Engineering 2006; 66:250-271. – reference: Farhat C, Lesoinne M, Pierson K. A scalable dual-primal domain decomposition method. Numerical Linear Algebra with Applications 2000; 7:687-714. – reference: Kim HH, Tu X. A three-level BDDC algorithm for mortar discretization. SIAM Journal on Numerical Analysis 2009; 47:1576-1600. – reference: Farhat C, Roux FX. An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems. SIAM Journal on Scientific and Statistical Computing 1992; 13:379-396. – reference: Tu X. BDDC algorithm for a mixed formulation of flows in porous media. Electronic Transactions on Numerical Analysis 2005; 20:164-179. – reference: Tu X. Three-level BDDC in two dimensions. International Journal for Numerical Methods in Engineering 2007; 69:33-59. – reference: Li J, Widlund OB. BDDC algorithms for incompressible Stokes equations. SIAM Journal on Numerical Analysis 2006; 44:2432-2455. – reference: Pavarino LF, Widlund OB, Zampini S. BDDC preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions. SIAM Journal on Scientific Computing 2010; 32:3604-3626. – reference: Li J, Widlund OB. On the use of inexact subdomain solvers for BDDC algorithms. Computer Methods in Applied Mechanics and Engineering 2007; 196:1415-1428. – reference: Farhat C. A Lagrange multiplier based divide and conquer finite element algorithm. Computing Systems in Engineering 1991; 2:149-156. – reference: Farhat C, Roux FX. A method of finite element tearing and interconnecting and its parallel solution algorithm. International Journal for Numerical Methods in Engineering 1991; 32:1205-1227. – reference: Dohrmann CR, Widlund OB. An overlapping Schwarz algorithm for almost incompressible elasticity. SIAM Journal on Numerical Analysis 2009; 47:2897-2923. – reference: Kim HH, Lee CO. A Neumann-Dirichlet preconditioner for a FETI-DP formulation of the two-dimensional Stokes problem with mortar methods. SIAM Journal on Scientific Computing 2006; 28:1133-1152. – reference: Kim HH, Lee CO, Park EH. A FETI-DP formulation for the Stokes problem without primal pressure components. SIAM Journal on Numerical Analysis 2010; 47:4142-4162. – reference: Li J. A dual-primal FETI method for incompressible Stokes equations. Numerische Mathematik 2005; 102:257-275. – reference: Braess D. Finite Elements, 2nd ed. Cambridge: Cambridge, UK, 2001. – reference: Bramble JH, Pasciak JE. A domain decomposition technique for Stokes problems. 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Snippet	SUMMARYA unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear... A unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear equations... SUMMARY A unified framework of dual-primal finite element tearing and interconnecting (FETI-DP) algorithms is proposed for solving the system of linear...
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SubjectTerms	Algorithms Approximation BDDC Computational methods in fluid dynamics Convergence Dirichlet problem domain decomposition Exact sciences and technology FETI-DP Finite element method Fluid dynamics Fluid flow Fundamental areas of phenomenology (including applications) incompressible Stokes Mathematical analysis Mathematics Methods of scientific computing (including symbolic computation, algebraic computation) Numerical analysis. Scientific computation Physics Sciences and techniques of general use Tearing
Title	A unified dual-primal finite element tearing and interconnecting approach for incompressible Stokes equations
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