INTERGRID TRANSFER OPERATORS AND MULTILEVEL PRECONDITIONERS FOR NONCONFORMING DISCRETIZATIONS

We discuss multilevel preconditioners of hierarchical basis and BPX ty pe for nonconforming discretizations of second and fourth order ellipt ic variational problems where the underlying subspace splitting of a n onconforming fine grid space is obtained from the natural sequence of nonconforming coarse grid spaces using appropriately designed intergri d transfer operators. We present a simple convergence theory which sho ws the importance of controlling the energy norm growth of the iterate d 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 no nconforming elements on uniform grids indicate boundedness of these no rms, with the exception of the Morley element where the condition numb ers deteriorate exponentially with the number of levels. The results c omplement recent work by Bramble, Pasciak and Xu, Brenner, Dorfler, an d the author.