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Super-localization of elliptic multiscale problems

Super-localization of elliptic multiscale problems

Authors :
Hauck, Moritz
Peterseim, Daniel
Publication Year :
2021

Abstract

Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a $d$-dimensional domain. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size parameter $H$. This holds even for under-resolved rough coefficients. However, the true challenge of numerical homogenization is the localized computation of a localized basis for such an operator-dependent approximate solution space. This paper presents a novel localization technique that enforces the super-exponential decay of the basis relative to $H$. This shows that basis functions with supports of width $\mathcal O(H|\log H|^{(d-1)/d})$ are sufficient to preserve the optimal algebraic rates of convergence in $H$ without pre-asymptotic effects. A sequence of numerical experiments illustrates the significance of the new localization technique when compared to the so far best localization to supports of width $\mathcal O(H|\log H|)$.<br />Comment: 22 pages, 7 figures

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2107.13211
Document Type :
Working Paper