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

We consider the problem of splitting a symmetric positive definite (SPD) stiffness matrix A arising from finite element discretization into the 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 a splitting into SPSD edge matrices and provide a feasible algorithm for their computation. 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...

• Type: Journal
• Year: 2006
• Where: COMPUTING
• Authors: J. K. Kraus, Josef Schicho
• Keywords: COMPUTING 2006, Edge Matrices, SPSD Edge Matrices, Symmetric Positive Definite