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This paper presents estimates of the convergence rate and complexity of analgebraic multilevel preconditioner based on piecewise constant coarse vectorspaces applied to the graph Laplacian. A bound is derived on the energy norm ofthe projection operator onto any piecewise constant vector space, which resultsin an estimate of the two-level convergence rate where the coarse level graphis obtained by matching. The two-level convergence of the method is then usedto establish the convergence of an Algebraic Multilevel Iteration that uses thetwo-level scheme recursively. On structured grids, the method is proven to haveconvergence rate $\approx (1-1/\log n)$ and $O(n\log n)$ complexity for eachcycle, where $n$ denotes the number of unknowns in the given problem. Numericalresults of the algorithm applied to various graph Laplacians are reported. Itis also shown that all the theoretical estimates derived for matching can begeneralized to the case of aggregates containing more than two vertices.

Algebraic multilevel preconditioners for the graph Laplacian based on matching in graphs