These notes are concerned with the numerical treatment of the coupling between second order elliptic problems that feature large contrast between their characteristic coefficients. In particular, we study the applicat...
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ISBN:
(纸本)9783642239144;9783642239137
These notes are concerned with the numerical treatment of the coupling between second order elliptic problems that feature large contrast between their characteristic coefficients. In particular, we study the application of Nitsche's method to set up a robust approximation of interface conditions in the framework of the finite element method. The notes are subdivided in three parts. Firstly, we review the weak enforcement of Dirichlet boundary conditions with particular attention to Nitsche's method and we discuss the extension of such technique to the coupling of Poisson equations. Secondly, we review the application of Nitsche's method to large contrast problems, discretised on computational meshes that capture the interface of discontinuity between coefficients. Finally, we extend the previous schemes to the case of unfitted meshes, which occurs when the computational mesh does not conform with the interface between subproblems.
This work is a follow up to previous articles of the same authors (Costaouec, Le Bris, and Lego II, Boletin Soc. Esp. Mat. Apl. 50:9-27, 2010;Blanc, Costaouec, Le Bris, and Lego II, Markov Processes and Related Fields...
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ISBN:
(纸本)9783642219429
This work is a follow up to previous articles of the same authors (Costaouec, Le Bris, and Lego II, Boletin Soc. Esp. Mat. Apl. 50:9-27, 2010;Blanc, Costaouec, Le Bris, and Lego II, Markov Processes and Related Fields, in press). It has been shown there, both numerically and theoretically, that the technique of antithetic variables successfully applies to stochastic homogenization of divergenceform linear elliptic problems and allows to reduce variance in computations. In (Costaouec, Le Bris, and Legoll, Boletin Soc. Esp. Mat. Apl. 50:9-27, 2010), variance reduction was assessed numerically for the diagonal terms of the homogenized matrix, in the case when the random field, that models uncertainty on some physical property at microscale, has a simple form. The numerical experiments have been complemented in Blanc, Costaouec, Le Bris, and Legoll (Markov Processes and Related Fields, in press) by a theoretical study. The main objective of this work is to proceed with some numerical experiments in a broader set of cases. We show the efficiency of the approach in each of the settings considered.
The paper describes techniques for constructing simplified models for problems governed by elliptic partial differential equations involving heterogeneous media. Examples of problems under consideration include electr...
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ISBN:
(纸本)9783642219429
The paper describes techniques for constructing simplified models for problems governed by elliptic partial differential equations involving heterogeneous media. Examples of problems under consideration include electro-statics and linear elasticity in composite materials, and flows in porous media. A common approach to such problems is to either up-scale the governing differential equation and then discretize the up-scaled equation, or to construct a discrete problem whose solution approximates the solution to the original problem under some constraints on the permissible loads. In contrast, the current paper suggests that it is in many situations advantageous to directly approximate the solution operator to the original differential equation. Such an approach has become feasible due to recent advances in numerical analysis, and can in a natural way handle situations that are challenging to existing techniques, such as those involving, e.g. concentrated loads, boundary effects, and irregular micro-structures. The capabilities of the proposed methodology are illustrated by numerical examples involving domains that are loaded on the boundary only, in which case the solution operator is a boundary integral operator such as, e.g., a Neumann-to-Dirichlet operator.
The purpose of this paper is to provide a mathematical analysis of the Adler-Wiser formula relating the macroscopic relative permittivity tensor to the microscopic structure of the crystal at the atomic level. The tec...
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ISBN:
(纸本)9783642219429
The purpose of this paper is to provide a mathematical analysis of the Adler-Wiser formula relating the macroscopic relative permittivity tensor to the microscopic structure of the crystal at the atomic level. The technical level of the presentation is kept at its minimum to emphasize the mathematical structure of the results. We also briefly review some models describing the electronic structure of finite systems, focusing on density operator based formulations, as well as the Hartree model for perfect crystals or crystals with a defect.
The fast multipole method (FMM) is a technique allowing the fast calculation of long-range interactions between N points in O(N) or O(N In N) steps with some prescribed error tolerance. The FMM has found many applicat...
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ISBN:
(纸本)9783642219429
The fast multipole method (FMM) is a technique allowing the fast calculation of long-range interactions between N points in O(N) or O(N In N) steps with some prescribed error tolerance. The FMM has found many applications in the field of integral equations and boundary element methods, in particular by accelerating the solution of dense linear systems arising from such formulations. Standard FMMs are derived from analytic expansions of the kernel, for example using spherical harmonics or Taylor expansions. In recent years, the range of applicability and the ease of use of FMMs has been extended by the introduction of black box (Fong and Darve, Journal of computational Physics 228:8712-8725, 2009) or kernel independent techniques (Ying, Biros and Zorin, Journal of computational Physics 196:591-626, 2004). In these approaches, the user only provides a subroutine to numerically calculate the interaction kernel. This allows changing the definition of the kernel with minimal change to the computer program. This paper presents a novel kernel independent FMM, which leads to diagonal multipole-to-local operators. This results in a significant reduction in the computational cost (Fong and Darve, Journal of computational Physics 228:8712-8725, 2009), in particular when high accuracy is needed. The approach is based on Cauchy's integral formula and the Laplace transform. We will present a short numerical analysis of the convergence and some preliminary numerical results in the case of a single level one dimensional FMM.
C-0 interior penalty methods are discontinuous Galerkin methods for fourth order problems. In this article we discuss various aspects of such methods including a priori error analysis, a posteriori error analysis and ...
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ISBN:
(纸本)9783642239144;9783642239137
C-0 interior penalty methods are discontinuous Galerkin methods for fourth order problems. In this article we discuss various aspects of such methods including a priori error analysis, a posteriori error analysis and fast solution techniques.
The 91st London Mathematical Society Durham Symposium took place from July 5th to 15th 2010, with more than 100 international participants attending. The Symposium focused on Numerical Analysis of Multiscale Problems ...
ISBN:
(数字)9783642220616
ISBN:
(纸本)9783642220609
The 91st London Mathematical Society Durham Symposium took place from July 5th to 15th 2010, with more than 100 international participants attending. The Symposium focused on Numerical Analysis of Multiscale Problems and this book contains 10 invited articles from some of the meeting's key speakers, covering a range of topics of contemporary interest in this area. Articles cover the analysis of forward and inverse PDE problems in heterogeneous media, high-frequency wave propagation, atomistic-continuum modeling and high-dimensional problems arising in modeling uncertainty. Novel upscaling and preconditioning techniques, as well as applications to turbulent multi-phase flow, and to problems of current interest in materials science are all addressed. As such this book presents the current state-of-the-art in the numerical analysis of multiscale problems and will be of interest to both practitioners and mathematicians working in those fields.
We study a few interesting issues that occur in multiscale modeling and computation for oscillatory dynamical systems that involve three or more separated scales. A new type of slow variables which do not formally hav...
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ISBN:
(纸本)9783642219429
We study a few interesting issues that occur in multiscale modeling and computation for oscillatory dynamical systems that involve three or more separated scales. A new type of slow variables which do not formally have bounded derivatives emerge from averaging in the fastest time scale. We present a few systems which have such new slow variables and discuss their characterization. The examples motivate a numerical multiscale algorithm that uses nested tiers of integrators which numerically solve the oscillatory system on different time scales. The communication between the scales follows the framework of the Heterogeneous Multiscale Method. The method's accuracy and efficiency are evaluated and its applicability is demonstrated by examples.
The proceedings represent the state of knowledge in the area of algorithmic differentiation (AD). The 31 contributed papers presented at the AD2012 conference cover the application of AD to many areas in science and e...
ISBN:
(数字)9783642300233
ISBN:
(纸本)9783642300226
The proceedings represent the state of knowledge in the area of algorithmic differentiation (AD). The 31 contributed papers presented at the AD2012 conference cover the application of AD to many areas in science and engineering as well as aspects of AD theory and its implementation in tools. For all papers the referees, selected from the program committee and the greater community, as well as the editors have emphasized accessibility of the presented ideas also to non-AD experts. In the AD tools arena new implementations are introduced covering, for example, Java and graphical modeling environments or join the set of existing tools for Fortran. New developments in AD algorithms target the efficiency of matrix-operation derivatives, detection and exploitation of sparsity, partial separability, the treatment of nonsmooth functions, and other high-level mathematical aspects of the numerical computations to be differentiated. Applications stem from the Earth sciences, nuclear engineering, fluid dynamics, and chemistry, to name just a few. In many cases the applications in a given area of science or engineering share characteristics that require specific approaches to enable AD capabilities or provide an opportunity for efficiency gains in the derivative computation. The description of these characteristics and of the techniques for successfully using AD should make the proceedings a valuable source of information for users of AD tools.
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