{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T20:15:57Z","timestamp":1773260157227,"version":"3.50.1"},"reference-count":47,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,10,1]]},"abstract":"Abstract<\/jats:title>\n We devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic problems that is capable of handling general meshes. The method hinges on discrete unknowns that are polynomials attached to the faces and cells of a coarse mesh; those attached to the cells can be eliminated locally using static condensation. The main building ingredient is a reconstruction operator, local to each coarse cell, that maps onto a fine-scale space spanned by oscillatory basis functions.\nThe present HHO method generalizes the ideas of some existing multiscale approaches, while providing the first complete analysis on general meshes. It also improves on those methods, taking advantage of the flexibility granted by the HHO framework. The method handles arbitrary orders of approximation \n \n \n \n k<\/m:mi>\n \u2265<\/m:mo>\n 0<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {k\\geq 0}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. For face unknowns that are polynomials of degree k<\/jats:italic>, we devise two versions of the method, depending on the polynomial degree \n \n \n \n (<\/m:mo>\n \n k<\/m:mi>\n -<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {(k-1)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> or k<\/jats:italic> of the cell unknowns. We prove, in the case of periodic coefficients, an energy-error estimate of the form \n \n \n \n (<\/m:mo>\n \n \n \u03b5<\/m:mi>\n \n 1<\/m:mn>\n 2<\/m:mn>\n <\/m:mfrac>\n <\/m:msup>\n +<\/m:mo>\n \n H<\/m:mi>\n \n k<\/m:mi>\n +<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n +<\/m:mo>\n \n \n (<\/m:mo>\n \n \u03b5<\/m:mi>\n H<\/m:mi>\n <\/m:mfrac>\n )<\/m:mo>\n <\/m:mrow>\n \n 1<\/m:mn>\n 2<\/m:mn>\n <\/m:mfrac>\n <\/m:msup>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {(\\varepsilon^{\\frac{1}{2}}+H^{k+1}+(\\frac{\\varepsilon}{H})^{\\frac{1}{2}})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, and we illustrate our theoretical findings on some test-cases.<\/jats:p>","DOI":"10.1515\/cmam-2018-0013","type":"journal-article","created":{"date-parts":[[2018,6,19]],"date-time":"2018-06-19T22:15:46Z","timestamp":1529446546000},"page":"723-748","source":"Crossref","is-referenced-by-count":19,"title":["A Hybrid High-Order Method for Highly Oscillatory Elliptic Problems"],"prefix":"10.1515","volume":"19","author":[{"given":"Matteo","family":"Cicuttin","sequence":"first","affiliation":[{"name":"Universit\u00e9 Paris-Est , CERMICS (ENPC) , 6\u20138 avenue Blaise Pascal, 77455 Marne-la-Vall\u00e9e Cedex 2; and Inria Paris, 2 rue Simone Iff, 75012 Paris , France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexandre","family":"Ern","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Paris-Est , CERMICS (ENPC) , 6\u20138 avenue Blaise Pascal, 77455 Marne-la-Vall\u00e9e Cedex 2; and Inria Paris, 2 rue Simone Iff, 75012 Paris , France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Simon","family":"Lemaire","sequence":"additional","affiliation":[{"name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, FSB-MATH-ANMC, Station 8, 1015 Lausanne, Switzerland; and Inria Lille \u2013 Nord Europe , 40 avenue Halley, 59650 Villeneuve d\u2019Ascq , France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2018,6,19]]},"reference":[{"key":"2023033110105317240_j_cmam-2018-0013_ref_001_w2aab3b7e1322b1b6b1ab2b2b1Aa","doi-asserted-by":"crossref","unstructured":"A. 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Vohral\u00edk,\nStable broken H1H^{1} and H\u2062(div)H(\\mathrm{div}) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions,\npreprint (2016), https:\/\/hal.inria.fr\/hal-01422204."},{"key":"2023033110105317240_j_cmam-2018-0013_ref_029_w2aab3b7e1322b1b6b1ab2b2c29Aa","doi-asserted-by":"crossref","unstructured":"D. Gilbarg and N. S. Trudinger,\nElliptic Partial Differential Equations of Second Order,\nClassics Math.,\nSpringer, Berlin, 2001.","DOI":"10.1007\/978-3-642-61798-0"},{"key":"2023033110105317240_j_cmam-2018-0013_ref_030_w2aab3b7e1322b1b6b1ab2b2c30Aa","doi-asserted-by":"crossref","unstructured":"V. Girault and P.-A. Raviart,\nFinite Element Methods for Navier\u2013Stokes Equations. Theory and Algorithms,\nSpringer Ser. Comput. 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Zhang,\nRemoving the cell resonance error in the multiscale finite element method via a Petrov\u2013Galerkin formulation,\nCommun. Math. Sci. 2 (2004), no. 2, 185\u2013205.","DOI":"10.4310\/CMS.2004.v2.n2.a3"},{"key":"2023033110105317240_j_cmam-2018-0013_ref_037_w2aab3b7e1322b1b6b1ab2b2c37Aa","doi-asserted-by":"crossref","unstructured":"V. V. Jikov, S. M. Kozlov and O. A. Oleinik,\nHomogenization of Differential Operators and Integral Functionals,\nSpringer, Berlin, 1994.","DOI":"10.1007\/978-3-642-84659-5"},{"key":"2023033110105317240_j_cmam-2018-0013_ref_038_w2aab3b7e1322b1b6b1ab2b2c38Aa","unstructured":"A. Konat\u00e9,\nM\u00e9thode multi-\u00e9chelle pour la simulation d\u2019\u00e9coulements miscibles en milieux poreux,\nPh.D. thesis, Universit\u00e9 Pierre et Marie Curie, 2017, https:\/\/tel.archives-ouvertes.fr\/tel-01558994."},{"key":"2023033110105317240_j_cmam-2018-0013_ref_039_w2aab3b7e1322b1b6b1ab2b2c39Aa","doi-asserted-by":"crossref","unstructured":"C. Le Bris, F. Legoll and A. Lozinski,\nMsFEM \u00e0 la Crouzeix\u2013Raviart for highly oscillatory elliptic problems,\nChin. Ann. Math. Ser. B 34 (2013), no. 1, 113\u2013138.","DOI":"10.1007\/s11401-012-0755-7"},{"key":"2023033110105317240_j_cmam-2018-0013_ref_040_w2aab3b7e1322b1b6b1ab2b2c40Aa","doi-asserted-by":"crossref","unstructured":"C. Le Bris, F. Legoll and A. Lozinski,\nAn MsFEM-type approach for perforated domains,\nSIAM Multiscale Model. Simul. 12 (2014), no. 3, 1046\u20131077.","DOI":"10.1137\/130927826"},{"key":"2023033110105317240_j_cmam-2018-0013_ref_041_w2aab3b7e1322b1b6b1ab2b2c41Aa","doi-asserted-by":"crossref","unstructured":"A. M\u00e5lqvist and D. Peterseim,\nLocalization of elliptic multiscale problems,\nMath. Comp. 83 (2014), 2583\u20132603.","DOI":"10.1090\/S0025-5718-2014-02868-8"},{"key":"2023033110105317240_j_cmam-2018-0013_ref_042_w2aab3b7e1322b1b6b1ab2b2c42Aa","doi-asserted-by":"crossref","unstructured":"L. Mu, J. Wang and X. Ye,\nA weak Galerkin generalized multiscale finite element method,\nJ. Comput. Appl. Math. 305 (2016), 68\u201381.","DOI":"10.1016\/j.cam.2016.03.017"},{"key":"2023033110105317240_j_cmam-2018-0013_ref_043_w2aab3b7e1322b1b6b1ab2b2c43Aa","doi-asserted-by":"crossref","unstructured":"D. Paredes, F. Valentin and H. M. Versieux,\nOn the robustness of multiscale hybrid-mixed methods,\nMath. Comp. 86 (2017), 525\u2013548.","DOI":"10.1090\/mcom\/3108"},{"key":"2023033110105317240_j_cmam-2018-0013_ref_044_w2aab3b7e1322b1b6b1ab2b2c44Aa","doi-asserted-by":"crossref","unstructured":"N. Sukumar and A. Tabarraei,\nConforming polygonal finite elements,\nInternat. J. Numer. Methods Engrg. 61 (2004), no. 12, 2045\u20132066.","DOI":"10.1002\/nme.1141"},{"key":"2023033110105317240_j_cmam-2018-0013_ref_045_w2aab3b7e1322b1b6b1ab2b2c45Aa","doi-asserted-by":"crossref","unstructured":"A. Veeser and R. Verf\u00fcrth,\nPoincar\u00e9 constants for finite element stars,\nIMA J. Numer. 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