Algebraic Two-Level Preconditioners for the Schur Complement Method

The solution of elliptic problems is challenging on parallel distributed memory computers since their Green's functions are global. To address this issue, we present a set of preconditioners for the Schur complement domain decomposition method. They implement a global coupling mechanism, through coarse-space components, similar to the one proposed in [Bramble, Pasciak, and Shatz, Math. Comp., 47 (1986), pp. 103--134]. The definition of the coarse-space components is algebraic; they are defined using the mesh partitioning information and simple interpolation operators. These preconditioners are implemented on distributed memory computers without introducing any new global synchronization in the preconditioned conjugate gradient iteration. The numerical and parallel scalability of those preconditioners are illustrated on two-dimensional model examples that have anisotropy and/or discontinuity phenomena.

[1]  J. Mandel,et al.  Convergence of a substructuring method with Lagrange multipliers , 1994 .

[2]  William Gropp,et al.  Domain Decomposition: Parallel Mul-tilevel Methods for Elliptic PDEs , 1996 .

[3]  P. Tallec.  Domain decomposition methods in computational mechanics , 1994 .

[4]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[5]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[6]  Tony F. Chan,et al.  An Energy-minimizing Interpolation for Robust Multigrid Methods , 1999, SIAM J. Sci. Comput..

[7]  L. Giraud,et al.  Schur complement preconditioners for anisotropic problems , 1999 .

[8]  Barry F. Smith,et al.  Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions , 1994 .

[9]  Tony F. Chan,et al.  The Interface Probing Technique in Domain Decomposition , 1992, SIAM J. Matrix Anal. Appl..

[10]  Alfio Quarteroni,et al.  Domain Decomposition Methods for Partial Differential Equations , 1999 .

[11]  MIXED FINITE ELEMENT SIMULATION OF HETEROJUNCTION STRUCTURES INCLUDING A BOUNDARY LAYER MODEL FOR THE QUASI‐FERMI LEVELS , 1994 .

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

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

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

[15]  P. Wesseling.  An Introduction to Multigrid Methods , 1992 .

[16]  Juan C. Meza,et al.  A Multigrid Preconditioner for the Semiconductor Equations , 1996, SIAM J. Sci. Comput..

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

[18]  T. Chan,et al.  Domain decomposition algorithms , 1994, Acta Numerica.

[19]  Iain S. Duff,et al.  MA27 -- A set of Fortran subroutines for solving sparse symmetric sets of linear equations , 1982 .

[20]  Barry F. Smith,et al.  Domain decomposition algorithms for the partial differential equations of linear elasticity , 1990 .

[21]  Ray S. TuminaroyAbstractWe.  Grid Transfer Operators for Highly Variable Coeecient Problems , 1993 .