{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T18:32:24Z","timestamp":1774722744196,"version":"3.50.1"},"reference-count":35,"publisher":"American Mathematical Society (AMS)","issue":"341","license":[{"start":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T00:00:00Z","timestamp":1700524800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["865751 RandomMultiScales"],"award-info":[{"award-number":["865751 RandomMultiScales"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["865751 RandomMultiScales"],"award-info":[{"award-number":["865751 RandomMultiScales"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["865751 RandomMultiScales"],"award-info":[{"award-number":["865751 RandomMultiScales"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["865751 RandomMultiScales"],"award-info":[{"award-number":["865751 RandomMultiScales"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n  <mml:semantics>\n    <mml:mi>d<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-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 <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H\">\n  <mml:semantics>\n    <mml:mi>H<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">H<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. 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 approximation space. This paper presents a novel localization technique that leads to a super-exponential decay of its basis relative to <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H\">\n  <mml:semantics>\n    <mml:mi>H<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">H<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. This suggests that basis functions with supports of width <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper O left-parenthesis upper H StartAbsoluteValue log upper H EndAbsoluteValue Superscript left-parenthesis d minus 1 right-parenthesis slash d Baseline right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>H<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo stretchy=\"false\">|<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>log<\/mml:mi>\n      <mml:mo>\u2061<\/mml:mo>\n      <mml:mi>H<\/mml:mi>\n      <mml:msup>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo stretchy=\"false\">|<\/mml:mo>\n        <\/mml:mrow>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo stretchy=\"false\">(<\/mml:mo>\n          <mml:mi>d<\/mml:mi>\n          <mml:mo>\u2212<\/mml:mo>\n          <mml:mn>1<\/mml:mn>\n          <mml:mo stretchy=\"false\">)<\/mml:mo>\n          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n            <mml:mo>\/<\/mml:mo>\n          <\/mml:mrow>\n          <mml:mi>d<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathcal O(H|\\log H|^{(d-1)\/d})<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> are sufficient to preserve the optimal algebraic rates of convergence in <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H\">\n  <mml:semantics>\n    <mml:mi>H<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">H<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> 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 <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper O left-parenthesis upper H StartAbsoluteValue log upper H EndAbsoluteValue right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>H<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo stretchy=\"false\">|<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>log<\/mml:mi>\n      <mml:mo>\u2061<\/mml:mo>\n      <mml:mi>H<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo stretchy=\"false\">|<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathcal O(H|\\log H|)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>.<\/p>","DOI":"10.1090\/mcom\/3798","type":"journal-article","created":{"date-parts":[[2022,10,13]],"date-time":"2022-10-13T12:22:20Z","timestamp":1665663740000},"page":"981-1003","source":"Crossref","is-referenced-by-count":30,"title":["Super-localization of elliptic multiscale problems"],"prefix":"10.1090","volume":"92","author":[{"given":"Moritz","family":"Hauck","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Peterseim","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,11,21]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492921000015","article-title":"Numerical homogenization beyond scale separation","volume":"30","author":"Altmann, Robert","year":"2021","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1081\/NFA-120039655","article-title":"Steklov eigenproblems and the representation of solutions of elliptic boundary value problems","volume":"25","author":"Auchmuty, Giles","year":"2004","journal-title":"Numer. 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