scholarly journals Convergence Rates for the Stratified Periodic Homogenization Problems

2016 ◽  
Vol 6 (8) ◽  
Author(s):  
Jie Zhao ◽  
Juan Wang
Keyword(s):  
Convergence Rates ◽  
Periodic Homogenization ◽  
Homogenization Problems

scholarly journals Kinetic Decomposition for Periodic Homogenization Problems

2009 ◽  
Vol 41 (1) ◽  
pp. 360-390 ◽  
Author(s):  
Pierre-Emmanuel Jabin ◽  
Athanasios E. Tzavaras
Keyword(s):  
Periodic Homogenization ◽  
Kinetic Decomposition ◽  
Homogenization Problems

scholarly journals COUPLING HETEROGENEOUS MULTISCALE FEM WITH RUNGE–KUTTA METHODS FOR PARABOLIC HOMOGENIZATION PROBLEMS: A FULLY DISCRETE SPACETIME ANALYSIS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250002 ◽  
Author(s):  
ASSYR ABDULLE ◽  
GILLES VILMART
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Convergence Rates ◽  
Implicit Methods ◽  
Fully Discrete ◽  
Discrete Analysis ◽  
Discrete Spacetime ◽  
Runge Kutta ◽  
Heterogeneous Multiscale ◽  
Homogenization Problems

Numerical methods for parabolic homogenization problems combining finite element methods (FEMs) in space with Runge–Kutta methods in time are proposed. The space discretization is based on the coupling of macro and micro finite element methods following the framework of the Heterogeneous Multiscale Method (HMM). We present a fully discrete analysis in both space and time. Our analysis relies on new (optimal) error bounds in the norms L2(H1), [Formula: see text], and [Formula: see text] for the fully discrete analysis in space. These bounds can then be used to derive fully discrete spacetime error estimates for a variety of Runge–Kutta methods, including implicit methods (e.g. Radau methods) and explicit stabilized method (e.g. Chebyshev methods). Numerical experiments confirm our theoretical convergence rates and illustrate the performance of the methods.

scholarly journals Combined effects of homogenization and singular perturbations: Quantitative estimates

2021 ◽  
pp. 1-34
Author(s):  
Weisheng Niu ◽  
Zhongwei Shen
Keyword(s):  
Large Scale ◽  
Convergence Rates ◽  
Elliptic Systems ◽  
Small Scale ◽  
Length Scale ◽  
Combined Effects ◽  
Periodic Homogenization ◽  
Quantitative Estimates ◽  
Lipschitz Estimate ◽  
Scale Estimate

We investigate quantitative estimates in periodic homogenization of second-order elliptic systems of elasticity with singular fourth-order perturbations. The convergence rates, which depend on the scale κ that represents the strength of the singular perturbation and on the length scale ε of the heterogeneities, are established. We also obtain the large-scale Lipschitz estimate, down to the scale ε and independent of κ. This large-scale estimate, when combined with small-scale estimates, yields the classical Lipschitz estimate that is uniform in both ε and κ.

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