Cbs Constants for Graph-laplacians and Application to Multilevel Methods for Discontinuous Galerkin Systems

The goal of this work is to derive and justify a multilevel preconditioner for symmetric discontinuous approximations of second order elliptic problems. Our approach is based on the following simple idea. The finite element space V of piece-wise polynomials of certain degree that are discontinuous on the partition T0 is projected onto the space of piece-wise constants on the same partition. This will constitute the finest space in the multilevel method. The projection of the discontinuous Galerkin system on this space is associated to the so-called “graph-Laplacian”. In 2-D this is a very simple M-matrix with −1 as off diagonal entries and current diagonal entries equal to the number of the neighbours through the interfaces of the current finite element. Then after consecutive aggregation of the finite elements we produce a sequence of spaces of piece-wise constant functions. We develop the concept of hierarchical splitting of the unknowns and using local analysis we derive uniform estimates for the constant in the strengthen Cauchy-BunyakowskiSchwarz (CBS) inequality. As a measure of the angle between the spaces of the splitting, this further is used to justify a multilevel preconditioner of the discontinuous Galerkin system in spirit of the work [4] of Axelsson and Vassilevski. key words: discontinuous Galerkin, second order elliptic equation, graph-Laplacian, multilevel preconditioning, CBS constant