An Aggregation Multilevel Method Using Smooth Error Vectors

Many algebraic multilevel methods for solving linear systems assume that the slowly converging, algebraically smooth error is locally constant. This assumption is often not true and can lead to poor performance of the method. Other multilevel methods require a description of the algebraically smooth error via knowledge of the near-nullspace of the operator, but this information may not always be available. This paper presents an aggregation multilevel method for problems where the near-nullspace of the operator is not known. The method uses samples of low-energy error vectors to construct its interpolation operator. The basis vectors for an aggregate are computed via a singular value decomposition of the sample vectors locally over that aggregate. Compared to many other methods that automatically adjust to the near-nullspace, this method does not require that the element stiffness matrices are available.

[1]  Thomas A. Manteuffel,et al.  Adaptive Smoothed Aggregation (αSA) , 2004, SIAM J. Sci. Comput..

[2]  Steve McCormick,et al.  Algebraic multigrid methods applied to problems in computational structural mechanics , 1989 .

[3]  Scott P. MacLachlan,et al.  Improving robustness in multiscale methods , 2004 .

[4]  J. Mandel,et al.  An Iterative Method with Convergence Rate Chosen a priori , 1999 .

[5]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[6]  Panayot S. Vassilevski,et al.  Element-Free AMGe: General Algorithms for Computing Interpolation Weights in AMG , 2001, SIAM J. Sci. Comput..

[7]  Jan Mandel,et al.  Iterative methods for p-version finite elements: preconditioning thin solids , 1996 .

[8]  Jacob Fish,et al.  Generalized Aggregation Multilevel solver , 1997 .

[9]  Jinchao Xu,et al.  Convergence estimates for multigrid algorithms without regularity assumptions , 1991 .

[10]  A. Brandt.  Multiscale Scientific Computation: Review 2001 , 2002 .

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

[12]  A. Brandt.  Algebraic multigrid theory: The symmetric case , 1986 .

[13]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[14]  V. E. Bulgakov,et al.  Iterative aggregation technique for large-scale finite element analysis of mechanical systems , 1994 .

[15]  StübenKlaus.  Algebraic multigrid (AMG) , 1983 .

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

[17]  T. Manteuffel,et al.  Adaptive Smoothed Aggregation ( α SA ) Multigrid ∗ , 2005 .

[18]  Petr Vaněk.  Acceleration of convergence of a two-level algorithm by smoothing transfer operators , 1992 .

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

[20]  Thomas A. Manteuffel,et al.  Algebraic Multigrid Based on Element Interpolation (AMGe) , 2000, SIAM J. Sci. Comput..

[21]  Thomas A. Manteuffel,et al.  Adaptive Smoothed Aggregation (AlphaSA) Multigrid , 2005, SIAM Rev..

[22]  Thomas A. Manteuffel,et al.  Adaptive Algebraic Multigrid , 2005, SIAM J. Sci. Comput..

[23]  Panayot S. Vassilevski,et al.  Spectral AMGe (ρAMGe) , 2003, SIAM J. Sci. Comput..

[24]  Edmond Chow,et al.  An unstructured multigrid method based on geometric smoothness , 2003, Numer. Linear Algebra Appl..