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A new technique for proving rate of convergence estimates of multigrid algorithms for symmetric positive definite problems will be given in this paper. The standard multigrid theory requires a" regularity and approximation" assumption. In contrast, the new theory requires only an easily verified approximation assumption. This leads to convergence results for multigrid refinement applications, problems with irregular coefficients, and problems whose coefficients have large jumps. In addition, the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications. References