Generalized Filtering Decomposition

This paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low frequency modes on convergence and so decrease or eliminate the plateau which is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process

Equidimensional modelling of flow and transport processes in fractured porous systems I

Flow and transport in fractured porous media play an important role for many environmental applications, e.g. the design of disposal systems for hazardous waste. The different hydraulic properties of the fractures and the surrounding rock matrix have a strong influence on the behaviour of the physical processes existing on site. In the two papers of this conference, we will present a new numerical concept to describe saturated flow and transport processes in arbitrarily fractured porous media. We will use an equidimensional approach where fracture and matrix are discretized with elements of the same dimension. To solve the problem, we developed a two-level multigrid method based on a hierarchical decomposition into a fracture problem and a matrix problem. This decoupled treatment of fracture and matrix allows us to handle the locally governing physical processes appropriately. In this paper we will also present convergence comparisons with classical multigrid and algebraic multigrid methods (AMG). In Neunhäuserer et al. (this issue, part II) we will discuss the effect of equidimensionality on the modelling results and the influence of the chosen transport discretisation technique

Wavelet Galerkin method for fractional elliptic differential equations

Under the guidance of the general theory developed for classical partial differential equations (PDEs), we investigate the Riesz bases of wavelets in the spaces where fractional PDEs usually work, and their applications in numerically solving fractional elliptic differential equations (FEDEs). The technique issues are solved and the detailed algorithm descriptions are provided. Compared with the ordinary Galerkin methods, the wavelet Galerkin method we propose for FEDEs has the striking benefit of efficiency, since the condition numbers of the corresponding stiffness matrixes are small and uniformly bounded; and the Toeplitz structure of the matrix still can be used to reduce cost. Numerical results and comparison with the ordinary Galerkin methods are presented to demonstrate the advantages of the wavelet Galerkin method we provide.Comment: 20 pages, 0 figure

Hierarchical decomposition of domains with fractures

We consider the efficient and robust numerical solution of elliptic problems with jumping coefficients occuring on a network of fractures. These thin geometric structures are resolved by anisotropic trapezoidal elements. We present an iterative solution concept based on a hierarchical separation of the fractures and the surrounding rock matrix. Upper estimates for the convergence rates are independent of the the jump of coefficients and of the width of the fractures and depend only polynomially on the number of refinement steps. The theoretical results are illustrated by numerical experiments

Water flow between soil aggregates

Aggregated soils are structured systems susceptible to non-uniform flow. The hydraulic properties depend on the aggregate fabric and the way the aggregates are assembled. We examined the hydraulic behavior of an aggregate packing. We focused on conditions when water mostly flows through the aggregates, leaving the inter-aggregate pore space air-filled. The aggregates were packed in 3mm thick slabs forming a quasi two-dimensional bedding. The larger aggregates were wetted with water and embedded in smaller aggregates equilibrated at a lower water content. The water exchange between wet and drier aggregates was monitored by neutron radiography. The three-dimensional arrangement of the aggregates was reconstructed by neutron tomography. The water flow turned out to be controlled by the contacts between aggregates, bottle-necks that slow down the flow. The bottle-neck effect is due to the narrow flow cross section of the contacts. The water exchange was simulated by considering the contact area between aggregates as the key parameter. In order to match the observed water flow, the contact area must be reduced by one to two orders of magnitude relative to that obtained from image analysis. The narrowness of the contacts is due to air-filled voids within the contact