{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T00:27:39Z","timestamp":1771979259485,"version":"3.50.1"},"reference-count":33,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"name":"Suomalainen Tiedeakatemia Foundation"},{"DOI":"10.13039\/501100002261","name":"RFBR","doi-asserted-by":"crossref","award":["14-01-00162\/15"],"award-info":[{"award-number":["14-01-00162\/15"]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The paper is concerned with sharp estimates of constants in the classical Poincar\u00e9\ninequalities and Poincar\u00e9-type inequalities for functions with zero mean values in a\nsimplicial domain or on a part of the boundary. These estimates are important for\nquantitative analysis of problems generated by differential equations where numerical\napproximations are typically constructed with the help of simplicial meshes. We suggest\neasily computable relations that provide sharp bounds of the respective constants and\ncompare these results with analytical estimates (if such estimates are known). In the\nlast section, we discuss possible applications and derive a computable majorant of the\ndifference between the exact solution of a boundary value problem and an arbitrary\nfinite dimensional approximation defined on a simplicial mesh.<\/jats:p>","DOI":"10.1515\/cmam-2015-0037","type":"journal-article","created":{"date-parts":[[2016,1,18]],"date-time":"2016-01-18T16:55:17Z","timestamp":1453136117000},"page":"277-298","source":"Crossref","is-referenced-by-count":10,"title":["Explicit Constants in Poincar\u00e9-Type Inequalities for Simplicial Domains and Application\nto A Posteriori Estimates"],"prefix":"10.1515","volume":"16","author":[{"given":"Svetlana","family":"Matculevich","sequence":"first","affiliation":[{"name":"Department of Mathematical Information Technology, University of Jyv\u00e4skyl\u00e4, P.O. Box 35, FI-40014 University of Jyv\u00e4skyl\u00e4, Finland"}]},{"given":"Sergey","family":"Repin","sequence":"additional","affiliation":[{"name":"V. A. Steklov Institute of Mathematics in St. Petersburg, Fontanka 27, 191011 St. Petersburg, Russia; and Department of Mathematical Information Technology, University of Jyv\u00e4skyl\u00e4, P.O. Box 35, FI-40014 University of Jyv\u00e4skyl\u00e4, Finland"}]}],"member":"374","published-online":{"date-parts":[[2016,1,16]]},"reference":[{"key":"ref31","first-page":"179","article-title":"Spectres et groupes cristallographiques Domaines euclidiens Invent no","volume":"58","author":"B\u00e9rard","year":"1980","journal-title":"Math"},{"key":"ref361","doi-asserted-by":"crossref","first-page":"191","DOI":"10.24033\/asens.510","article-title":"Sur les probl\u00e8mes fondamentaux de la physique math\u00e9matique Ann","volume":"3","author":"Steklov","year":"1902","journal-title":"Sci \u00c9c Norm Sup\u00e9r"},{"key":"ref81","first-page":"289","article-title":"Eigenvalue comparison theorems and its geometric applications Math no","volume":"143","author":"Cheng","year":"1975","journal-title":"Z"},{"key":"ref311","first-page":"57","article-title":"Sur les \u00e9quations de la physique math\u00e9matique Rend Mat Palermo Explicit Constants in Poincar\u00e9 - Type Inequalities A posteriori error estimation for variational problems with uniformly convex functionals Math no","volume":"8","author":"Poincar\u00e9","year":"1894","journal-title":"Comput"},{"key":"ref61","first-page":"290","article-title":"Guaranteed lower bounds for eigenvalues Math no","volume":"83","author":"Carstensen","year":"2014","journal-title":"Comp"},{"key":"ref351","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/S0045-7825(96)01136-X","article-title":"A posteriori error estimation for elastoplastic problems based on duality theory Comput no","volume":"138","author":"Repin","year":"1996","journal-title":"Methods Appl Mech Engrg"},{"key":"ref231","first-page":"2659","article-title":"A posteriori error estimates for time - dependent reaction - di usion problems based on the Payne Weinberger inequality Discrete Contin no","volume":"35","author":"Matculevich","year":"2015","journal-title":"Syst"},{"key":"ref221","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-23099-8","article-title":"eds ) Automated solution of di erential equations by the finite element method Lect Springer Heidelberg","volume":"84","author":"Logg","year":"2012","journal-title":"Notes Comput Sci Eng"},{"key":"ref261","first-page":"165","article-title":"A guaranteed bound of the optimal constant in the error estimates for linear triangular element in : Topics in Numerical Analysis Comput Springer Vienna","volume":"15","author":"Nakao","year":"2001","journal-title":"Suppl"},{"key":"ref321","article-title":"A Posteriori Estimates for Partial Di erential Equations Radon Ser De Gruyter Berlin","volume":"4","author":"Repin","year":"2008","journal-title":"Comput Appl Math"},{"key":"ref141","doi-asserted-by":"crossref","first-page":"2484","DOI":"10.1137\/070688675","article-title":"An analysis of a FETI - DP algorithm on irregular subdomains in the plane SIAM no","volume":"46","author":"Klawonn","year":"2008","journal-title":"Numer Anal"},{"key":"ref171","first-page":"9","article-title":"The legacy of Vladimir Andreevich Steklov Notices Amer no","volume":"61","author":"Kuznetsov","year":"2014","journal-title":"Math Soc"},{"key":"ref191","doi-asserted-by":"crossref","first-page":"112903","DOI":"10.1063\/1.3246834","article-title":"Maximizing Neumann fundamental tones of triangles no","volume":"50","author":"Laugesen","year":"2009","journal-title":"Math Phys"},{"key":"ref331","doi-asserted-by":"crossref","first-page":"874","DOI":"10.1007\/s10958-009-9363-9","article-title":"Estimates of deviations from exact solutions of variational inequalities based upon Payne Weinberger inequal - ity no","volume":"157","author":"Repin","year":"2009","journal-title":"Math Sci"},{"key":"ref211","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1007\/s13160-013-0120-6","article-title":"Guaranteed high - precision estimation for interpolation constants on triangular finite elements Jpn no","volume":"30","author":"Liu","year":"2013","journal-title":"Ind Appl Math"},{"key":"ref11","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1137\/0713021","article-title":"On the angle condition in the finite element method SIAM no","volume":"13","author":"Babu\u0161ka","year":"1976","journal-title":"Numer Anal"},{"key":"ref281","first-page":"286","article-title":"An optimal Poincar\u00e9 inequality for convex domains Arch Rational Mech","volume":"5","author":"Payne","year":"1960","journal-title":"Anal"},{"key":"ref101","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1137\/070685841","article-title":"Domain decomposition for less regular subdomains : Overlapping Schwarz in two dimensions SIAM no","volume":"46","author":"Dohrmann","year":"2008","journal-title":"Numer Anal"},{"key":"ref131","first-page":"93","article-title":"A ne Weyl groups and the boundary value eigenvalue problems of the Laplacian Interdis - cip no","volume":"16","author":"Hoshikawa","year":"2010","journal-title":"Inform Sci"},{"key":"ref291","first-page":"819","article-title":"The eigenvalues of an equilateral triangle SIAM no","volume":"11","author":"Pinsky","year":"1980","journal-title":"Math Anal"},{"key":"ref161","first-page":"2049","article-title":"On the two - dimensional sloshing problem Proc Math no","volume":"460","author":"Kozlov","year":"2004","journal-title":"Lond Phys Eng Sci"},{"key":"ref181","first-page":"353","article-title":"Functional a posteriori error estimates for parabolic time - periodic boundary value problems Comput no","volume":"15","author":"Langer","year":"2015","journal-title":"Methods Appl Math"},{"key":"ref301","doi-asserted-by":"crossref","first-page":"211","DOI":"10.2307\/2369620","article-title":"Sur les \u00e9quations aux d\u00e9riv\u00e9es partielles de la physique math\u00e9matique Amer no","volume":"12","author":"Poincar\u00e9","year":"1890","journal-title":"Math"},{"key":"ref151","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1002\/mma.442","article-title":"The ice - fishing problem : The fundamental sloshing frequency versus geometry of holes Math no","volume":"27","author":"Kozlov","year":"2004","journal-title":"Methods Appl Sci"},{"key":"ref111","doi-asserted-by":"crossref","first-page":"668","DOI":"10.1007\/BF00948809","article-title":"Sloshing frequencies no","volume":"34","author":"Fox","year":"1983","journal-title":"Angew Math Phys"},{"key":"ref71","first-page":"691","article-title":"A posteriori error analysis for elliptic PDEs on domains with complicated structures Numer no","volume":"96","author":"Carstensen","year":"2004","journal-title":"Math"},{"key":"ref51","first-page":"1465","article-title":"Fully reliable localized error control in the FEM SIAM no","volume":"21","author":"Carstensen","year":"2000","journal-title":"Sci Comput"},{"key":"ref371","article-title":"Domain Decomposition Methods Algorithms and Theory Springer Ser Springer Berlin","volume":"34","author":"Toselli","year":"2005","journal-title":"Comput Math"},{"key":"ref21","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1090\/trans2\/231\/04","article-title":"Eigenvalue inequalities for mixed Steklov problems in : Operator Theory and its Applications Amer American Mathematical Society Providence","author":"Ba\u00f1uelos","year":"2010","journal-title":"Math Soc Transl Ser"},{"key":"ref91","article-title":"The Finite Element Method for Elliptic Problems Stud North - Holland Amsterdam","volume":"4","author":"Ciarlet","year":"1978","journal-title":"Math Appl"},{"key":"ref271","doi-asserted-by":"crossref","first-page":"3195","DOI":"10.1002\/mma.3290","article-title":"Exact constants in Poincar\u00e9 type inequalities for functions with zero mean boundary traces Math no","volume":"38","author":"Nazarov","year":"2014","journal-title":"Methods Appl Sci"},{"key":"ref241","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1080\/1024123021000053664","article-title":"Eigenstructure of the equilateral triangle II The Neumann problem Math no","volume":"8","author":"McCartin","year":"2002","journal-title":"Probl Eng"},{"key":"ref341","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/978-3-319-10705-9_21","article-title":"Estimates of constants in boundary - mean trace inequalities and applications to error analysis in : Numerical Mathematics and Advanced Applications Lect Springer Cham","volume":"103","author":"Repin","year":"2013","journal-title":"Notes Comput Sci Eng"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/16\/2\/article-p277.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0037\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0037\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T22:59:52Z","timestamp":1680303592000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0037\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1,16]]},"references-count":33,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2016,2,20]]},"published-print":{"date-parts":[[2016,4,1]]}},"alternative-id":["10.1515\/cmam-2015-0037"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2015-0037","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1,16]]}}}