Intergrid transfer operators and multilevel preconditioners for nonconforming discretizations Original Research Article

We discuss multilevel preconditioners of hierarchical basis and BPX type for nonconforming discretizations of second and fourth order elliptic variational problems where the underlying subspace splitting of a nonconforming fine grid space is obtained from the natural sequence of nonconforming coarse grid spaces using appropriately designed intergrid transfer operators. We present a simple convergence theory which shows the importance of controlling the energy norm growth of the iterated coarse-to-fine-grid operators. It enters both upper and lower bounds for the condition number of the preconditioned linear system, and can be checked numerically in the case of regular dyadic refinement. For the standard sets of intergrid transfer operators (prolongations based on nodal value averaging), the numerical tests with some low order nonconforming elements on uniform grids indicate boundedness of these norms, with the exception of the Morley element where the condition numbers deteriorate exponentially with the number of levels. The results complement recent work by Bramble, Pasciak and Xu, Brenner, Dörfler, and the author.