Multigrid transfers for nonsymmetric systems based on Schur complements and Galerkin projections

SUMMARY A framework is proposed for constructing algebraic multigrid transfer operators suitable for nonsymmetric positive definite linear systems. This framework follows a Schur complement perspective as this is suitable for both symmetric and nonsymmetric systems. In particular, a connection between algebraic multigrid and approximate block factorizations is explored. This connection demonstrates that the convergence rate of a two-level model multigrid iteration is completely governed by how well the coarse discretization approximates a Schur complement operator. The new grid transfer algorithm is then based on computing a Schur complement but restricting the solution space of the corresponding grid transfers in a Galerkin-style so that a far less expensive approximation is obtained. The final algorithm corresponds to a Richardson-type iteration that is used to improve a simple initial prolongator or a simple initial restrictor. Numerical results are presented illustrating the performance of the resulting algebraic multigrid method on highly nonsymmetric systems. Copyright © 2013 John Wiley & Sons, Ltd.

[1]  Thomas A. Manteuffel,et al.  Towards Adaptive Smoothed Aggregation (AlphaSA) for Nonsymmetric Problems , 2010, SIAM J. Sci. Comput..

[2]  Alain Dervieux,et al.  Unstructured multigridding by volume agglomeration: Current status , 1992 .

[3]  Marian Brezina,et al.  Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems , 2005, Computing.

[4]  Randolph E. Bank,et al.  Kernel Preserving Multigrid Methods for Convection-Diffusion Equations , 2005, SIAM J. Matrix Anal. Appl..

[5]  P. Wesseling,et al.  Geometric multigrid with applications to computational fluid dynamics , 2001 .

[6]  Petr Vanek,et al.  An Aggregation Multigrid Solver for convection-diffusion problems onunstructured meshes. , 1998 .

[7]  P. M. De Zeeuw,et al.  Matrix-dependent prolongations and restrictions in a blackbox multigrid solver , 1990 .

[8]  Achi Brandt,et al.  Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, Revised Edition , 2011 .

[9]  Jacob Fish,et al.  An algebraic two‐level preconditioner for asymmetric, positive‐definite systems , 2001 .

[10]  Marian Brezina,et al.  Energy Optimization of Algebraic Multigrid Bases , 1998, Computing.

[11]  J. W. Ruge,et al.  4. Algebraic Multigrid , 1987 .

[12]  Donald B. Bliss,et al.  Wake Generation Compressibility Effects in Unsteady Aerodynamics , 1999 .

[13]  O. Axelsson,et al.  Algebraic multilevel preconditioning methods, II , 1990 .

[14]  Robert D. Falgout,et al.  Compatible Relaxation and Coarsening in Algebraic Multigrid , 2009, SIAM J. Sci. Comput..

[15]  Jonathan J. Hu,et al.  A new smoothed aggregation multigrid method for anisotropic problems , 2007, Numer. Linear Algebra Appl..

[16]  Ray S. Tuminaro,et al.  A New Petrov--Galerkin Smoothed Aggregation Preconditioner for Nonsymmetric Linear Systems , 2008, SIAM J. Sci. Comput..

[17]  O. Axelsson.  Iterative solution methods , 1995 .

[18]  Yvan Notay,et al.  Algebraic multigrid and algebraic multilevel methods: a theoretical comparison , 2005, Numer. Linear Algebra Appl..

[19]  W. Wall,et al.  Truly monolithic algebraic multigrid for fluid–structure interaction , 2011 .

[20]  Marian Brezina,et al.  Convergence of algebraic multigrid based on smoothed aggregation , 1998, Numerische Mathematik.

[21]  Jacob B. Schroder,et al.  A General Interpolation Strategy for Algebraic Multigrid Using Energy Minimization , 2011, SIAM J. Sci. Comput..

[22]  Achi Brandt,et al.  Inadequacy of first-order upwind difference schemes for some recirculating flows , 1991 .

[23]  J. Mandel,et al.  Energy optimization of algebraic multigrid bases , 1999 .

[24]  Achi Brandt,et al.  Distributed Relaxation Multigrid and Defect Correction Applied to the Compressible Navier-Stokes Equations , 1999 .

[25]  Achi Brandt,et al.  Accelerated Multigrid Convergence and High-Reynolds Recirculating Flows , 1993, SIAM J. Sci. Comput..

[26]  Dimitri J. Mavriplis,et al.  Directional Agglomeration Multigrid Techniques for High-Reynolds Number Viscous Flows , 1998 .

[27]  Wolfgang A. Wall,et al.  An algebraic variational multiscale-multigrid method based on plain aggregation for convection-diffusion problems , 2009 .

[28]  Michel Fortin,et al.  An algebraic multilevel parallelizable preconditioner for large-scale CFD problems , 1997 .

[29]  Yvan Notay,et al.  Algebraic analysis of two‐grid methods: The nonsymmetric case , 2010, Numer. Linear Algebra Appl..

[30]  William L. Briggs,et al.  A multigrid tutorial, Second Edition , 2000 .

[31]  P. Vassilevski,et al.  Algebraic multilevel preconditioning methods. I , 1989 .

[32]  Artem Napov,et al.  An Algebraic Multigrid Method with Guaranteed Convergence Rate , 2012, SIAM J. Sci. Comput..