2,775 research outputs found
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
Robust Discretization of Flow in Fractured Porous Media
Flow in fractured porous media represents a challenge for discretization
methods due to the disparate scales and complex geometry. Herein we propose a
new discretization, based on the mixed finite element method and mortar
methods. Our formulation is novel in that it employs the normal fluxes as the
mortar variable within the mixed finite element framework, resulting in a
formulation that couples the flow in the fractures with the surrounding domain
with a strong notion of mass conservation. The proposed discretization handles
complex, non-matching grids, and allows for fracture intersections and
termination in a natural way, as well as spatially varying apertures. The
discretization is applicable to both two and three spatial dimensions. A priori
analysis shows the method to be optimally convergent with respect to the chosen
mixed finite element spaces, which is sustained by numerical examples
Functional Analysis and Exterior Calculus on Mixed-Dimensional Geometries
We are interested in differential forms on mixed-dimensional geometries, in
the sense of a domain containing sets of -dimensional manifolds, structured
hierarchically so that each -dimensional manifold is contained in the
boundary of one or more dimensional manifolds.
On any given -dimensional manifold, we then consider differential
operators tangent to the manifold as well as discrete differential operators
(jumps) normal to the manifold. The combined action of these operators leads to
the notion of a semi-discrete differential operator coupling manifolds of
different dimensions. We refer to the resulting systems of equations as
mixed-dimensional, which have become a popular modeling technique for physical
applications including fractured and composite materials.
We establish analytical tools in the mixed-dimensional setting, including
suitable inner products, differential and codifferential operators, Poincar\'e
lemma, and Poincar\'e--Friedrichs inequality. The manuscript is concluded by
defining the mixed-dimensional minimization problem corresponding to the
Hodge-Laplacian, and we show that this minimization problem is well-posed
Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
In this paper, we discuss a general multiscale model reduction framework
based on multiscale finite element methods. We give a brief overview of related
multiscale methods. Due to page limitations, the overview focuses on a few
related methods and is not intended to be comprehensive. We present a general
adaptive multiscale model reduction framework, the Generalized Multiscale
Finite Element Method. Besides the method's basic outline, we discuss some
important ingredients needed for the method's success. We also discuss several
applications. The proposed method allows performing local model reduction in
the presence of high contrast and no scale separation
The LifeV library: engineering mathematics beyond the proof of concept
LifeV is a library for the finite element (FE) solution of partial
differential equations in one, two, and three dimensions. It is written in C++
and designed to run on diverse parallel architectures, including cloud and high
performance computing facilities. In spite of its academic research nature,
meaning a library for the development and testing of new methods, one
distinguishing feature of LifeV is its use on real world problems and it is
intended to provide a tool for many engineering applications. It has been
actually used in computational hemodynamics, including cardiac mechanics and
fluid-structure interaction problems, in porous media, ice sheets dynamics for
both forward and inverse problems. In this paper we give a short overview of
the features of LifeV and its coding paradigms on simple problems. The main
focus is on the parallel environment which is mainly driven by domain
decomposition methods and based on external libraries such as MPI, the Trilinos
project, HDF5 and ParMetis.
Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar
Schnelle Löser für Partielle Differentialgleichungen
The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds
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