The Analysis of Multigrid Algorithms with Nonnested Spaces or Noninherited Quadratic Forms

We provide a theory for the analysis of multigrid algorithms for symmetric positive definite problems with nonnested spaces and noninherited quadratic forms. By this we mean that the form on the coarser grids need not be related to that on the finest, i.e., we do not stay within the standard variational setting. In this more general setting, we give new estimates corresponding to the <!-- MATH $\mathcal{V}$ --> cycle, <!-- MATH $\mathcal{W}$ --> cycle and a <!-- MATH $\mathcal{V}$ --> cycle algorithm with a variable number of smoothings on each level. In addition, our algorithms involve the use of nonsymmetric smoothers in a novel way.