Geodesic
Multiscale analysis of linear evolution equations with applications to nonlocal models for heterogeneous media
Du, Qiang1  ; Lipton, Robert2 ; Mengesha, Tadele3
1 Department of Applied Physics and Applied Mathematics, Columbia University, New York, 10027, USA.
2 Department of Mathematics, Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA.
3 Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 5, pp. 1425-1455

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The method of two scale convergence is implemented to study the homogenization of time-dependent nonlocal continuum models of heterogeneous media. Two integro-differential models are considered: the nonlocal convection-diffusion equation and the state-based peridynamic model in nonlocal continuum mechanics. The asymptotic analysis delivers both homogenized dynamics as well as strong approximations expressed in terms of a suitable corrector theory. The method provides a natural analog to that for the time-dependent local PDE models with highly oscillatory coefficients with the distinction that the driving operators considered in this work are bounded.

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DOI : 10.1051/m2an/2015080
Classification : 74Q05, 74E05, 74H10, 45F99, 45P05
Keywords: Multiscale analysis, peridynamics, nonlocal equations, Navier equation, homogenization, heterogeneous materials, two-scale convergence

Du, Qiang 1 ; Lipton, Robert 2 ; Mengesha, Tadele 3

1 Department of Applied Physics and Applied Mathematics, Columbia University, New York, 10027, USA.
2 Department of Mathematics, Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA.
3 Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA. 
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     title = {Multiscale analysis of linear evolution equations with applications to nonlocal models for heterogeneous media},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1425--1455},
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Du, Qiang; Lipton, Robert; Mengesha, Tadele. Multiscale analysis of linear evolution equations with applications to nonlocal models for heterogeneous media. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 5, pp. 1425-1455. doi: 10.1051/m2an/2015080

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