A priori and a posteriori error analysis for numerical homogenization: a unified framework

Numerical methods coupling macroscopic and microscopic solvers for the efficient solution of partial differential equations with multiple scales have attracted much attention these last few years. Efficient implementations of such methods need a thorough understanding of the subtle interplay between the macroscopic scale (often the scale of interest) and the microscopic scale taking into account the fine-scale behavior of the problem of interest. This paper is concerned with the error analysis of a class of numerical homogenization methods, the so-called heterogeneous multiscale finite element methods. We discuss recent results obtained for the a priori and the a posteriori error analysis of these numerical methods and give a general framework to perform such analyses.

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