Convergence analysis of inexact two-grid methods: Multigrid cycles
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Multigrid is a popular iterative solver for a large class of linear systems that arise from discretized partial differential equations. Typically, it is a recursive call of two-grid procedure and hence can be treated as an inexact two-grid scheme. In this paper, we present a systematic convergence analysis of standard multigrid methods based on the inexact two-grid theory developed by Xu and Zhang (2020). Two alternating combinations of the V-cycle and W-cycle multigrid methods are also analyzed. More specifically, we establish new upper bounds for the convergence factor of multigrid methods in a purely algebraic manner. Moreover, our analysis allows the coarsest-grid problem to be solved approximately.
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