, Volume 77, Issue 1, pp 57–75 | Cite as

Algebraic Multigrid Based on Computational Molecules, 1: Scalar Elliptic Problems

  • J. K. KrausEmail author
  • J. Schicho


We consider the problem of splitting a symmetric positive definite (SPD) stiffness matrix A arising from finite element discretization into a sum of edge matrices thereby assuming that A is given as the sum of symmetric positive semidefinite (SPSD) element matrices. We give necessary and sufficient conditions for the existence of an exact splitting into SPSD edge matrices and address the problem of best positive (nonnegative) approximation.

Based on this disassembling process we present a new concept of ``strong'' and ``weak'' connections (edges), which provides a basis for selecting the coarse-grid nodes in algebraic multigrid methods. Furthermore, we examine the utilization of computational molecules (small collections of edge matrices) for deriving interpolation rules. The reproduction of edge matrices on coarse levels offers the opportunity to combine classical coarsening algorithms with effective (energy minimizing) interpolation principles yielding a flexible and robust new variant of AMG.

AMS Subject Classiffications

65F10 65N20 65N30 


Edge matrices algebraic multigrid interpolation weights coarse-grid selection 


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Copyright information

© Springer-Verlag Wien 2006

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied MathematicsLinzAustria

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