Algebraic Multigrid Based on Computational Molecules, 2: Linear Elasticity Problems

This paper deals with a new approach in algebraic multigrid (AMG) for self-adjoint and elliptic problems arising from finite-element discretization of linear elasticity problems. Generalizing our approach for scalar problems [J. K. Kraus and J. Schicho, Computing, 77 (2006), pp. 57-75], we propose an edge-matrix concept for point-block systems of linear algebraic equations. This gives a simple and reliable method for the evaluation of the strength of nodal dependence that can be applied to symmetric positive definite non-M matrices. In this paper the edge-matrix concept is developed for and applied to (two- as well as three-dimensional) linear elasticity problems. We consider approximate splittings of the problem-related element matrices into symmetric positive semidefinite edge matrices of rank one. The reproduction of edge matrices on coarse levels offers the opportunity to combine classical coarsening techniques with robust (energy-minimizing) interpolation schemes: The “computational molecules” involved in this process are assembled from edge matrices. This yields a flexible new variant of AMG that allows also for an efficient solution of problems with discontinuous coefficients, e.g., for composite materials.

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