Volume 44, pp. 472-496, 2015.

Algebraic distance for anisotropic diffusion problems: multilevel results

Achi Brandt, James Brannick, Karsten Kahl, and Irene Livshits

Abstract

In this paper, we motivate, discuss the implementation, and present the resulting numerics for a new definition of strength of connection which uses the notion of algebraic distance as defined originally in the bootstrap algebraic multigrid framework (BAMG). We use this algebraic distance measure together with compatible relaxation and least-squares interpolation to derive an algorithm for choosing suitable coarse grids and accurate interpolation operators for algebraic multigrid algorithms. The main tool of the proposed strength measure is the least-squares functional defined by using a set of test vectors that in general is computed using the bootstrap process. The motivating application is the anisotropic diffusion problem, in particular, with non-grid aligned anisotropy. We demonstrate numerically that the measure yields a robust technique for determining strength of connectivity among variables for both two-grid and multigrid bootstrap algebraic multigrid methods. The proposed algebraic distance measure can also be used in any other coarsening procedure assuming that a rich enough set of near-kernel components of the matrix for the targeted system is known or is computed as in the bootstrap process.

Full Text (PDF) [5.8 MB]

Key words

bootstrap algebraic multigrid, least-squares interpolation, algebraic distances, strength of connection

AMS subject classifications

65N55, 65N22 , 65F10

Links to the cited ETNA articles

[2]Vol. 10 (2000), pp. 1-20 Achi Brandt: General highly accurate algebraic coarsening

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