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Analysis of the heterogeneous multiscale method for elliptic homogenization problems. (English) Zbl 1060.65118
The heterogeneous multiscale method (HMM) is a general methodology for designing sublinear algorithms by expliting scale separation and other special features of the problem. It consists of two components: selection of a macroscopic solver and estimating the missing macroscale data by solving locally the fine scale problem. This paper estimates the error between the numerical solutions of HMM and the solutions of the problem $-\text{div} (A(x)\nabla U(x))=f(x)$, $x\in D$, $U(x)=0$, $x\in\partial D$. The authors prove a general statement that this error is controlled by a standard error in the macroscale solver plus a new term due to the error in estimating the stiffness matrix. This second part is only done for either periodic or random homogenisation problems.

##### MSC:
 65N15 Error bounds (BVP of PDE) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65C30 Stochastic differential and integral equations 35B27 Homogenization; equations in media with periodic structure (PDE) 35J25 Second order elliptic equations, boundary value problems
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