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

AbstractThe 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

Mathematical Modeling, Numerical Techniques, and Computer Simulation of Flows and Transport in Porous Media

this paper we present a variety of models in groundwater hydrology that have been used in computer simulation for design of remediation and clean-up technologies. We also discuss the important question of the choice of the approximation method for the corresponding mathematical problem. In fluid reservoirs (aquifer and petroleum reservoirs) there are two imperative practical requirements: the method should conserve the mass locally and should produce accurate velocities (fluxes) even for highly nonhomogeneous media with large jumps in the physical properties. This is the reason that the finite volume method with harmonic averaging of the coefficients has been very popular and successful in computer simulation of flows in porous media. However, when the problem requires accurate description of the topography and the hydrological structure, a more general technique based on the finite element approximation is needed. The mixed finite element method has these properties. Since its introduction by Raviart and Thomas [23] and its implementation by Ewing and Wheeler [17] for flow problems, it has become a standard way of deriving highorder conservative approximations. It should be noted that the lowest-order mixed method realized on rectangles (or parallelepipeds) with certain numerical integration produces cell-centered finite differences with harmonic averaging. In Section 5 we describe briefly the mixed finite element method for the linearized pressure equation using Raviart-Thomas finite elements. This will lead to a symmetric but indefinite system for the unknown pressure and velocity (flux). Next, we discuss briefly the algorithms involved in the solution of this saddle point type problem and introduce a variant of the classical Uzawa method. This variant was studied re..