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Algebraic distance for anisotropic diffusion problems: multilevel results

In this paper we motivate, discuss the implementation and present the resulting numerics for a new definition of strength of connection which is based on the notion of algebraic distance. This algebraic distance measure, combined with compatible relaxation, is used to choose suitable coarse grids and accurate interpolation operators for algebraic multigrid algorithms. The main tool of the proposed measure is the least squares functional defined using a set of relaxed test vectors. The motivating application is the anisotropic diffusion problem, in particular problems 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 solvers. %We illustrate the use of the measure to construct, in addition, an adaptive aggregation form of interpolation for the targeted anisotropic problems. %Our approach is not a two-level approach -- we provide preliminary results that show its extendability to multigrid. The proposed algebraic distance measure can also be used in any other coarsening procedure, assuming a rich enough set of the near-kernel components of the matrix for the targeted system is known or computed.