fifteen years ago, bramble, pasciak and xu developed a framework for analyzing the multigrid method (the analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms, mathematics of computation, 1991.) it was, known as bpx framework, widely used in the analysis of mutligrid and domain decomposition methods. the framework was extended several times, covering less regularity or non-symmetric, and other cases. however an apparent limit of the framework is that it could not incorporate the number of smoothings in the v-cycle analysis. therefore, the framework is limited in nonnested multigrid methods (as one-smoothing multigrid methods won\'t converge for almost all nonnested cases), and it produces variable v-cycle, relaxed coarse-level correction, or non-uniform convergence rate v-cycle methods, or other non-optimal results in analysis thus far. therefore, most non-nested finite element problems were show to converge only for w-cycles based bpx framework, or an even earlier frame work of bank and dupont. this paper completes a long time effort in extending the bpx framework so that the number of smoothings is included in the v-cycle analysis. we will apply the extended bpx framework to the analysis of many v-cycle nonnested multigrid methods. some of them were previously proven for two-level and w-cycle iterations only.