This paper reviews the variational multiscale stabilization of standard finite elementmethods for linear partial differential equations that exhibitmultiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor L2 approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering.
CITATION STYLE
Peterseim, D. (2016). Variational multiscale stabilization and the exponential decay of fine-scale correctors. In Lecture Notes in Computational Science and Engineering (Vol. 114, pp. 341–367). Springer Verlag. https://doi.org/10.1007/978-3-319-41640-3_11
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