CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems

The goal of this work is to derive and justify a multilevel preconditioner of optimal arithmetic complexity for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. Our approach is based on the following simple idea given in [R.D. Lazarov, P.S. Vassilevski, L.T. Zikatanov, Multilevel preconditioning of second order elliptic discontinuous Galerkin problems, Preprint, 2005]. The finite element space V of piece-wise polynomials, discontinuous on the partition T, is projected onto the space of piece-wise constant functions on the same partition that constitutes the largest space in the multilevel method. The discontinuous Galerkin finite element system on this space is associated to the so-called “graph-Laplacian”. In 2-D this is a sparse M-matrix with -1 as off diagonal entries and nonnegative row sums. Under the assumption that the finest partition is a result of multilevel refinement of a given coarse mesh, we develop the concept of hierarchical splitting of the unknowns. Then using local analysis we derive estimates for the constants in the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality, which are uniform with respect to the levels. This measure of the angle between the spaces of the splitting was used by Axelsson and Vassilevski in [Algebraic multilevel preconditioning methods II, SIAM J. Numer. Anal. 27 (1990) 1569–1590] to construct an algebraic multilevel iteration (AMLI) for finite element systems. The main contribution in this paper is a construction of a splitting that produces new estimates for the CBS constant for graph-Laplacian. As a result we have a preconditioner for the system of the discontinuous Galerkin finite element method of optimal arithmetic complexity.