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Asymptotic analysis for periodic structures
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TLDR
In this article, the authors give a systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate.Abstract:
This is a reprinting of a book originally published in 1978. At that time it was the first book on the subject of homogenization, which is the asymptotic analysis of partial differential equations with rapidly oscillating coefficients, and as such it sets the stage for what problems to consider and what methods to use, including probabilistic methods. At the time the book was written the use of asymptotic expansions with multiple scales was new, especially their use as a theoretical tool, combined with energy methods and the construction of test functions for analysis with weak convergence methods. Before this book, multiple scale methods were primarily used for non-linear oscillation problems in the applied mathematics community, not for analyzing spatial oscillations as in homogenization. In the current printing a number of minor corrections have been made, and the bibliography was significantly expanded to include some of the most important recent references. This book gives systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate. The book continues to be interesting and useful to readers of different backgrounds, both from pure and applied mathematics, because of its informal style of introducing the multiple scale methodology and the detailed proofs.read more
Citations
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Homogenization of a neutronic critical diffusion problem with drift
TL;DR: In this paper, the homogenization of an eigenvalue problem for a cooperative system of weakly coupled elliptic partial differential equations, called the neutronic multigroup diffusion model, in a periodic heterogeneous domain is studied.
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Homogenized Dynamics of Stochastic Partial Differential Equations with Dynamical Boundary Conditions
Wei Wang,Jinqiao Duan +1 more
TL;DR: In this article, a homogenized macroscopic model for a microscopic heterogeneous stochastic system under random influence is derived, where the randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium.
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Machine computation using the exponentially convergent multiscale spectral generalized finite element method
TL;DR: It is proved that the MS-GFEM method has an exponential rate of convergence and can be applied to the solution of very large FE systems associated with the discrete solution of elliptic PDE.
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Numerical homogenization of the absolute permeability using the conformal-nodal and mixed-hybrid finite element method
Wouter Zijl,Anna Trykozko +1 more
TL;DR: In this article, the authors presented the theory of upscaling from a physical point of view aiming at understanding, rather than mathematical rigorousness, for steady single-phase flow of a fluid with constant viscosity and density.
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A Method for Efficient Calculation of Diffusion and Reactions of Lipophilic Compounds in Complex Cell Geometry
TL;DR: A mathematical model is developed that addresses the cellular behaviour of toxic foreign compounds and can be extended and refined to include more reactants, and/or more complex reaction chains, enzyme distribution etc, and is suitable for modelling cellular metabolism involving membrane partitioning also at higher levels of complexity.
References
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Measurable eigenvectors for Hermitian matrix-valued polynomials☆
TL;DR: In this article, a polynomial Hermitian matrices over the complex number field C are considered and the relative eigenvalue problem is formulated as a linear combination of the repeated eigenvalues of the matrix A(p)x = λEx, x ϵ Cm.