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In this paper, we study a posteriori error estimators which aid multileveliterative solvers for linear systems with graph Laplacians. In earlier workssuch estimates were computed by solving global optimization problems, whichcould be computationally expensive. We propose a novel strategy to computethese estimates by constructing a Helmholtz decomposition on the graph based ona spanning tree and the corresponding cycle space. To compute the errorestimator, we solve efficiently the linear system on the spanning tree, andthen we solve approximately a least-squares problem on the cycle space. As weshow, such an estimator has a nearly-linear computational complexity for sparsegraphs under certain assumptions. Numerical experiments are presented todemonstrate the efficacy of the proposed method.

A Posteriori Error Estimates for Solving Graph Laplacians