Stochastic homogenization of elliptic equations

2020 ◽  
pp. 131-154
Author(s):  
Charles Smart
Keyword(s):  
Elliptic Equations ◽  
Stochastic Homogenization

scholarly journals Special quasirandom structures: A selection approach for stochastic homogenization

2016 ◽  
Vol 22 (1) ◽  
Author(s):  
Claude Le Bris ◽  
Frédéric Legoll ◽  
William Minvielle
Keyword(s):  
Solid State ◽  
Elliptic Equations ◽  
Variance Reduction ◽  
Divergence Form ◽  
Atomistic Simulations ◽  
Stochastic Homogenization ◽  
Selection Approach ◽  
Reduction Approach

AbstractWe adapt and study a variance reduction approach for the homogenization of elliptic equations in divergence form. The approach, borrowed from atomistic simulations and solid-state science

scholarly journals Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems

2021 ◽  
Author(s):  
Julian Fischer ◽  
Stefan Neukamm
Keyword(s):  
Error Estimates ◽  
Elliptic Equations ◽  
Regularity Theory ◽  
Optimal Order ◽  
Small Scale ◽  
Elliptic Pdes ◽  
Stochastic Homogenization ◽  
Elliptic Case ◽  
Nonlinear Elliptic ◽  
Spatially Homogeneous

AbstractWe derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on $$\mathbb {R}^d$$ R d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale $$\varepsilon >0$$ ε > 0 , we establish homogenization error estimates of the order $$\varepsilon $$ ε in case $$d\geqq 3$$ d ≧ 3 , and of the order $$\varepsilon |\log \varepsilon |^{1/2}$$ ε | log ε | 1 / 2 in case $$d=2$$ d = 2 . Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence $$\varepsilon ^\delta $$ ε δ . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order $$(L/\varepsilon )^{-d/2}$$ ( L / ε ) - d / 2 for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) $$C^{1,\alpha }$$ C 1 , α regularity theory is available.

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