Svetozar Margenov
Department of Information Technology
Institute for Parallel Processing
Bulgarian Academy of Sciences
Sofia, Bulgaria
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. The finite element space of piece-wise polynomials, discontinuous on the considered partition, 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 a local analysis we derive estimates for the constants in the strengthen Cauchy-Bunyakowski-Schwarz (CBS) inequality for graph-Laplacians.