We study a generalized Crank-Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k 2 + h 2 , where k and h are the parameters for the time and space meshes, respectively.

We consider an initial-boundary value problem for ∂tu − ∂ −α t ∇ 2 u = f (t), that is, for a fractional diffusion (−1 < α < 0) or wave (0 < α < 1) equation. A numerical solution is found by applying a piecewise-linear, discontinuous Galerkin method in time combined with a piecewise-linear, conforming finite element method in space. The time mesh is graded appropriately near t = 0, but the spatial mesh is quasiuniform. Previously, we proved that the error, measured in the spatial L 2 -norm, is of order k 2+α − + h 2 ℓ(k), uniformly in t, where k is the maximum time step, h is the maximum diameter of the spatial finite elements, α − = min(α, 0) ≤ 0 and ℓ(k) = max(1, | log k|). Here, we generalize a known result for the classical heat equation (i.e., the case α = 0) by showing that at each time level tn the solution is superconvergent with respect to k: the error is of order (k 3+2α − + h 2 )ℓ(k). Moreover, a simple postprocessing step employing Lagrange interpolation yields a superconvergent approximation for any t. Numerical experiments indicate that our theoretical error bound is pessimistic if α < 0. Ignoring logarithmic factors, we observe that the error in the DG solution at t = tn, and after postprocessing at all t, is of order k 3+α − + h 2 .

We propose and analyze a time-stepping discontinuous Petrov-Galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time-fractional subdiffusion problems. We prove the existence, uniqueness, and stability of approximate solutions and derive error estimates. To achieve high order convergence rates from the time discretizations, the time mesh is graded appropriately near t = 0 to compensate for the singular (temporal) behavior of the exact solution near t = 0 caused by the weakly singular kernel, but the spatial mesh is quasi uniform. In the L∞((0, T ); L 2 (Ω))-norm, ((0, T ) is the time domain and Ω is the spatial domain); for sufficiently graded time meshes, a global convergence of order k m+α/2 + h r+1 is shown, where 0 < α < 1 is the fractional exponent, k is the maximum time step, h is the maximum diameter of the elements of the spatial mesh, and m and r are the degrees of approximate solutions in time and spatial variables, respectively. Numerical experiments indicate that our theoretical error bound is pessimistic. We observe that the error is of order k m+1 + h r+1 , that is, optimal in both variables.

We employ a piecewise-constant, discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Denoting the maximum time step by k, we prove an a priori error bound of order k under realistic assumptions on the regularity of the solution. We also show that a spatial discretization using continuous, piecewise-linear finite elements leads to an additional error term of order h 2 max(1, log k −1 ). Some simple numerical examples illustrate this convergence behaviour in practice.

We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that the method is first order accurate at the nth time level t n , but the error bound includes a factor t −1 n if we assume no smoothness of the initial data. We also show that for smoother initial data the growth in the error bound as t n decreases is milder, and in some cases absent altogether. Our error bounds generalize known results for the classical heat equation and are illustrated for a model problem.

The numerical solution for a class of sub-diffusion equations involving a parameter in the range −1 < α < 0 is studied. For the time discretization, we use an implicit finite-difference Crank-Nicolson method and show that the error is of order k 2+α , where k denotes the maximum time step. A nonuniform time step is employed to compensate for the singular behaviour of the exact solution at t = 0. We also consider a fully discrete scheme obtained by applying linear finite elements in space to the proposed time-stepping scheme. We prove that the additional error is of order h 2 max(1, log k −1), where h is the parameter for the space mesh. Numerical experiments on some sample problems demonstrate our theoretical result.

Natural fractured media are highly unpredictable because of existing complex structures at the fracture and at the network levels. Fractures are by themselves heterogeneous objects of broadly distributed sizes, shapes, and orientations that are interconnected in large correlated networks. With little field data and evidence, numerical modeling can provide important information on the underground hydraulic phenomena. However, it must overcome several barriers. First, the complex network structure produces a structure difficult to mesh. Second, the absence of a priori homogenization scale, along with the double fracture and network heterogeneity levels, requires the calculation of large but finely resolved fracture networks resulting in very large simulation domains. To tackle these two related issues, we reduce the highly complex geometry of the fractures by applying a local transformation that suppresses the cumbersome meshing configurations while keeping the networks fundamental, geological, and geometrical characteristics. We show that the flow properties are marginally affected while the problem complexity (i.e., memory capacity and resolution time) can be divided by orders of magnitude. The goal of this article is to propose a method of resolution which takes into account the geometrical complexity met in the networks and which makes it possible to treat a few thousand fractures. The principal aim of this article is to present a tool to slowly modify the structures of the fracture networks to have a good quality mesh with a marginal loss in precision. Introduction.The underground waste repository projects and exploitation of hot, dry, rock geothermal energy have spurred studies in fracture networks transport properties. Many site-specific studies have flourished around various projects [7], which are generally based on a careful characterization of the structure of the fractured rock mass, with the classical difficulty of deducing three-dimensional information from one-or two-dimensional field data [3]. Then the hydraulic properties are computed by using reconstructed model networks, based on the experimental geometrical characteristics, and various flow models. The flow fluid modeling in underground media requires taking into account the very high geological heterogeneity. The complexity of the geological mediums comes from metamorphic and sedimentary processes, and mechanics which create structures having hydraulic properties varying on several orders of magnitude and correlated on a broad range of scales. A general overview of stochastic generation of fractured media problems is given in [11]. Numerically, the challenge is to integrate this range of heterogeneity in models of great extension and good resolution outcome, with linear systems from 10 6 to 10 8 unknown factors. In the fractured medium with a matrix of very low permeability, the fluid flow is focused in the highly heterogeneous fractures [21]. The objective of numerical modeling consists of simulating hydraulic phenomena in a large number of ne...

Time-stepping hp-versions discontinuous Galerkin (DG) methods for the numerical solution of fractional subdiffusion problems of order −α with −1 < α < 0 will be proposed and analyzed. Generic hp-version error estimates are derived after proving the stability of the approximate solution. For h-version DG approximations on appropriate graded meshes near t = 0, we prove that the error is of order O(k max{2,p}+ α 2 ), where k is the maximum time-step size and p ≥ 1 is the uniform degree of the DG solution. For hp-version DG approximations, by employing geometrically refined time-steps and linearly increasing approximation orders, exponential rates of convergence in the number of temporal degrees of freedom are shown. Finally, some numerical tests are given.