A Coarse Space Construction Based on Local Dirichlet-to-Neumann Maps

Coarse-grid correction is a key ingredient of scalable domain decomposition methods. In this work we construct coarse-grid space using the low-frequency modes of the subdomain Dirichlet-to-Neumann maps and apply the obtained two-level preconditioners to the extended or the original linear system arising from an overlapping domain decomposition. Our method is suitable for parallel implementation, and its efficiency is demonstrated by numerical examples on problems with large heterogeneities for both manual and automatic partitionings.

[1]  O. Widlund,et al.  Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions , 1994 .

[2]  Owe Axelsson,et al.  Generalized Augmented Matrix Preconditioning Approach and its Application to Iterative Solution of Ill-Conditioned Algebraic Systems , 2000, SIAM J. Matrix Anal. Appl..

[3]  M. Gander,et al.  Why Restricted Additive Schwarz Converges Faster than Additive Schwarz , 2003 .

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

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

[6]  Frédéric Nataf,et al.  Optimal Interface Conditions for Domain Decomposition Methods , 1994 .

[7]  J. Meijerink,et al.  An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients , 1999 .

[8]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[9]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[10]  Robert Scheichl,et al.  Scaling up through domain decomposition , 2009 .

[11]  Martin J. Gander,et al.  Optimized Schwarz Methods without Overlap for the Helmholtz Equation , 2002, SIAM J. Sci. Comput..

[12]  Robert Scheichl,et al.  Analysis of FETI methods for multiscale PDEs , 2008, Numerische Mathematik.

[13]  Reinhard Nabben,et al.  Deflation and Balancing Preconditioners for Krylov Subspace Methods Applied to Nonsymmetric Matrices , 2008, SIAM J. Matrix Anal. Appl..

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

[15]  Reinhard Nabben,et al.  Algebraic Multilevel Krylov Methods , 2009, SIAM J. Sci. Comput..

[16]  J. Meijerink,et al.  The construction of projection vectors for a deflated ICCG method applied to problems with extreme contrasts in the coefficients , 2001 .

[17]  Cornelis Vuik,et al.  Comparison of Two-Level Preconditioners Derived from Deflation, Domain Decomposition and Multigrid Methods , 2009, J. Sci. Comput..

[18]  Frédéric Magoulès,et al.  Algebraic approximation of Dirichlet-to-Neumann maps for the equations of linear elasticity , 2006 .

[19]  Xiao-Chuan Cai,et al.  A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems , 1999, SIAM J. Sci. Comput..

[20]  Zi-Cai Li,et al.  Schwarz Alternating Method , 1998 .

[21]  J. Mandel.  Balancing domain decomposition , 1993 .

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

[23]  Martin J. Gander,et al.  Optimized Multiplicative, Additive, and Restricted Additive Schwarz Preconditioning , 2007, SIAM J. Sci. Comput..

[24]  Cornelis Vuik,et al.  A Comparison of Two-Level Preconditioners Based on Multigrid and Deflation , 2010, SIAM J. Matrix Anal. Appl..

[25]  José F. Escobar.  The Geometry of the First Non-zero Stekloff Eigenvalue , 1997 .

[26]  J. Pasciak,et al.  The Construction of Preconditioners for Elliptic Problems by Substructuring. , 2010 .

[27]  R. Nicolaides.  Deflation of conjugate gradients with applications to boundary value problems , 1987 .

[28]  Marian Brezina,et al.  Balancing domain decomposition for problems with large jumps in coefficients , 1996, Math. Comput..