Convergence Rates of Solutions for Elliptic Reiterated Homogenization Problems

2020 ◽  
Vol 51 (3) ◽  
pp. 839-856
Author(s):  
Juan Wang ◽  
Jie Zhao
Keyword(s):  
Convergence Rates ◽  
Reiterated Homogenization ◽  
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 Convergence Rates in L 2 for Elliptic Homogenization Problems

2011 ◽  
Vol 203 (3) ◽  
pp. 1009-1036 ◽  
Author(s):  
Carlos E. Kenig ◽  
Fanghua Lin ◽  
Zhongwei Shen
Keyword(s):  
Convergence Rates ◽  
Homogenization Problems

scholarly journals Homogenization Problem in a Domain with Double Oscillating Boundary

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Jie Zhao ◽  
Juan Wang
Keyword(s):  
Convergence Rate ◽  
Boundary Data ◽  
Taylor Expansion ◽  
Periodic Boundary ◽  
Convergence Rates ◽  
Homogenization Problem ◽  
Convergence Of Solutions ◽  
The Poisson Equation ◽  
Oscillating Boundary ◽  
Homogenization Problems

In this paper, we study the convergence of solutions for homogenization problems about the Poisson equation in a domain with double oscillating locally periodic boundary. Such a problem arises in the processing of devices with very small features. We utilize second-order Taylor expansion of boundary data in combination with boundary correctors to obtain the convergence rate in H1-norm. This work explores the domain with double oscillating boundary and also shows the influence of the amplitudes and periods of the oscillations to convergence rates of solutions.

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