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A numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined function spaces a well posed 3D-1D problem is obtained from the original fully 3D problem and the solution is then found by a PDE-constrained optimization reformulation. This is a domain decomposition strategy in which unknown interface variables are introduced and a suitably defined cost functional, expressing the error in fulfilling interface conditions, is minimized constrained by the constitutive equations on the subdomains. The resulting discrete problem is robust with respect to geometrical complexity thanks to the use of independent discretizations on the various subdomains. Meshes of different sizes can be used without affecting the conditioning of the discrete linear system, and this is a peculiar aspect of the considered formulation. An efficient solving strategy is further proposed, based on the use of a gradient based solver and yielding a method ready for parallel implementation. A numerical experiment on a problem with known analytical solution shows the accuracy of the method, and two examples on more complex configurations are proposed to address the applicability of the approach to practical problems. [ABSTRACT FROM AUTHOR]

Coupled 3D-1D problems arise in many practical applications, in an attempt to reduce the computational burden in simulations where cylindrical inclusions with a small section are embedded in a much larger domain. Nonetheless the resolution of such problems can be non trivial, both from a mathematical and a geometrical standpoint. Indeed 3D-1D coupling requires to operate in non standard function spaces, and, also, simulation geometries can be complex for the presence of multiple intersecting domains. Recently, a PDE-constrained optimization based formulation has been proposed for such problems, proving a well posed mathematical formulation and allowing for the use of non conforming meshes for the discrete problem. Here an unconstrained optimization formulation of the problem is derived and an efficient gradient based solver is proposed for such formulation, along with a suitable preconditioner to speed up the iterative solver. Some numerical tests on quite complex configurations are discussed to show the viability of the method. [ABSTRACT FROM AUTHOR]

A new discretization approach is presented for the simulation of flow in complex poro-fractured media described by means of the Discrete Fracture and Matrix Model. The method is based on the numerical optimization of a properly defined cost-functional and allows to solve the problem without any constraint on mesh generation, thus overcoming one of the main complexities related to efficient and effective simulations in realistic DFMs. [ABSTRACT FROM AUTHOR]

In this note the issue of fluid flow computation in a Discrete Fracture-Matrix (DFM) model is addressed. In such a model, a network of percolative fractures delimits porous matrix blocks. Two frameworks are proposed for the coupling between the two media. First, a FEM-BEM technique is considered, in which finite elements on non-conforming grids are used on the fractures, whereas a boundary element method is used on the blocks; the coupling is pursued by a PDE-constrained optimization formulation of the problem. Second, a VEM-VEM technique is considered, in which a 2D and a 3D virtual element method are used on the fractures and on the blocks, respectively, taking advantage of the flexibility of VEM in using arbitrary meshes in order to ease the meshing process and the consequent enforcement of the matching conditions on fractures and blocks. [ABSTRACT FROM AUTHOR]

The present work deals with the simulation of the flow in Discrete Fracture Networks (DFN), using the mixed formulation of the Virtual Element Method (VEM) on polygonal conforming meshes. The flexibility of the VEM in handling polygonal meshes is used to easily generate a conforming mesh even in the case of intricate DFNs. Mixed Virtual Elements of arbitrary polynomial accuracy are then used for the discretization of the velocity field. The well posedness of the resulting discrete problem is shown. Numerical results on simple problems are proposed to show convergence properties of the method with respect to known analytic solutions, whereas some tests on fairly complex networks are also reported showing its applicability and effectiveness. [ABSTRACT FROM AUTHOR]

In this paper we propose a modified construction for the polynomial basis on polygons used in the Virtual Element Method (VEM). This construction is alternative to the usual monomial basis used in the classical construction of the VEM and is designed in order to improve numerical stability. For badly shaped elements the construction of the projection matrices required for assembling the local coefficients of the linear system within the VEM discretization of Partial Differential Equations can result very ill conditioned. The proposed approach can be easily implemented within an existing VEM code in order to reduce the possible ill conditioning of the elemental projection matrices. Numerical results applied to an hydro-geological flow simulation that often produces very badly shaped elements show a clear improvement of the quality of the numerical solution, confirming the viability of the approach. The method can be conveniently combined with a classical implementation of the VEM and applied element-wise, thus requiring a rather moderate additional numerical cost. [ABSTRACT FROM AUTHOR]

Among the major challenges in performing underground flow simulations in fractured media are geometrical complexities in the domain and uncertainty in the problem parameters, including the geometrical configuration. The Discrete Fracture Network (DFN) model is largely applied in order to properly account for the directionality of the flow in fractured media. Generation of DFN configurations is usually based on stochastic data and this contributes to generate very complex geometrical configurations for which a conforming mesh generation is often infeasible. Moreover, uncertainty in the geometrical and hydro-geological properties calls for a deep uncertainty quantification analysis; the corresponding huge computational cost of the simulations requires modern efficient approaches faster and cheaper than the classical Monte Carlo approach. In this paper we numerically investigate both these aspects, proposing a viable solution for dealing with geometrical complexities arising in the computation of the hydraulic head and in the solution of the unsteady transport problem of a passive scalar in the DFN, and for dealing with uncertainties in hydro-geological parameters of the fracture distribution considered. [ABSTRACT FROM AUTHOR]

Explicit generators for high-order ( r>1) scalar and vector finite element spaces generally used in numerical electromagnetism are presented and classical degrees of freedom, the so-called moments, revisited. Properties of these generators on simplicial meshes are investigated, and a general technique to restore duality between moments and generators is proposed. Algebraic and exponential optimal h- and r-error rates are numerically validated for high-order edge elements on the problem of Maxwell's eigenvalues in a square domain. [ABSTRACT FROM AUTHOR]

We consider a variational model for the joint total variation filtering (TV) and the segmentation in the optic flow estimation. The model is based on a functional with a spatially varying regularization parameter to solve such an ill-posed problem. We present an adaptive approach based on a posteriori error indicators which allows us to select locally the optimal values of the diffusion coefficient in the functional. We show that the adaptive process applied to the linear part of the functional (with respect to the optic flow variable) fulfills the segmentation objective. Moreover, this adaptive approach provides an approximation of the Mumford-Shah functional in the sense of the $$\Gamma $$ -convergence of a family of discrete energies. The simultaneous filtering and segmentation are achieved within this approach with accuracy and a reduced number of degrees of freedom, which improves each task to obtain a reliable optic flow estimation. We present some numerical simulations to show the performances of the method for the segmentation and the simultaneous segmentation-filtering. [ABSTRACT FROM AUTHOR]

This article shows in detail how to construct in a simple and ordered way a set of rational functions defined on the pyramid topology. The set of functions is parameterized by an integer p . It is shown that these functions, defined in a hierarchical way, constitute a basis for a complete polynomial interpolation space of degree p on the pyramid domain. In order to help this definition we use a denumerable sequence of orthogonal polynomials defined on an interval of the real line. A priori, any increasing sequence of polynomials in one variable can be used. The bases are constructed to be used in the class C 0 method of finite elements. The rational functions thus defined can be combined to represent any polynomial of degree p . Thus, given an arbitrary number p , one defines a finite element whose geometry is a pyramid that has associated a complete interpolation space of degree p . Moreover, this element is adequate to be used with the p -adaptive technique on heterogeneous meshes of finite elements hierarchical. [ABSTRACT FROM AUTHOR]

This article deals with the analysis of an iterative non-overlapping domain decomposition (DD) method for elliptic problems, using Robin-type boundary condition on the inter-subdomain boundaries, which can be solved in parallel with local communications. The proposed iterative method allows us to relax the continuity condition for Lagrange multipliers on the inter-subdomain boundaries. In order to derive the corresponding discrete problem, we apply a non-conforming Galerkin method using lowest order Crouzeix-Raviart elements. The convergence of the iterative scheme is obtained by proving that the spectral radius of the matrix associated with the fixed point iterations is less than 1. Parallel computations have been carried out and the numerical experiments confirm the theoretical results established in this paper. [ABSTRACT FROM AUTHOR]

In a companion paper (Bastian et al. 2007, this issue) we introduced an abstract definition of a parallel and adaptive hierarchical grid for scientific computing. Based on this definition we derive an efficient interface specification as a set of C++ classes. This interface separates the applications from the grid data structures. Thus, user implementations become independent of the underlying grid implementation. Modern C++ template techniques are used to provide an interface implementation without big performance losses. The implementation is realized as part of the software environment DUNE (). Numerical tests demonstrate the flexibility and the efficiency of our approach. [ABSTRACT FROM AUTHOR]

The here presented GLAciology NUmerical SImulation Toolbox (GLANUSIT) is a software application which provides a user friendly environment for the numerical simulation of large ice masses evolution. The graphical user interface has been developed in MATLAB while the core of GLANUSIT contains the original FORTRAN codes, which develop the specific numerical methods for the solution of the complex shallow ice model. This highly nonlinear model governs the coupled thermodynamical and hydrodynamical processes. The global algorithm mainly consists on a fixed point iteration between the different subproblems. The numerical solution of each subproblem requires specific techniques, which are not common in present software packages, as for example the part of moving boundaries solvers included in the code. Finally, a practical case study with real data is presented. [Copyright &y& Elsevier]

This paper presents an object-oriented programming approach for the design of numerical homogenization programs, called JHomogenizer. It currently includes five functional modules to compute effective permeability and simple codes for computing solutions for flow in porous media. Examples with graphical output are shown to illustrate some functionalities of the program. A series of numerical examples demonstrates the effectiveness of the methodology for two-phase flow in heterogeneous reservoirs. The software is freely available, and the open architecture of the program facilitates further development and can adapt to suit specific needs easily and quickly. [ABSTRACT FROM AUTHOR]

We analyze the error in the p version of the of the finite element method when the effect of the quadrature error is taken in the load vector. We briefly study some results on the H norm error and present some new results for the error in the L norm. We investigate the quadrature error due to the numerical integration of the right hand side We present theoretical and computational examples showing the sharpness of our results. [ABSTRACT FROM AUTHOR]

We analyze the error in thep version of the of the finite element method when the effect of the quadrature error is taken in the load vector. We briefly study some results on theH1 norm error and present some new results for the error in theL2 norm. We investigate the quadrature error due to the numerical integration of the right hand side We present theoretical and computational examples showing the sharpness of our results. [ABSTRACT FROM AUTHOR]