{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:42:31Z","timestamp":1776789751928,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"266","license":[{"start":{"date-parts":[[2009,11,13]],"date-time":"2009-11-13T00:00:00Z","timestamp":1258070400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    A multiplicative Schwarz overlapping domain decomposition method is considered for solving elliptic boundary value problems. By equipping the relevant Sobolev spaces on the subdomains with wavelet bases, adaptive wavelet methods are used for approximately solving the subdomain problems. The union of the wavelet bases forms a frame for the Sobolev space on the domain as a whole. The resulting method is proven to be optimal in the sense that, in linear complexity, the iterands converge with the same rate as the sequence over\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N element-of double-struck upper N\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">N<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">N \\in \\mathbb {N}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the best approximation from the span of the best\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n                        <mml:semantics>\n                          <mml:mi>N<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    frame elements. Numerical results are given for the method applied to Poisson\u2019s equation.\n                  <\/p>","DOI":"10.1090\/s0025-5718-08-02186-8","type":"journal-article","created":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T13:09:23Z","timestamp":1259672963000},"page":"619-644","source":"Crossref","is-referenced-by-count":6,"title":["A multiplicative Schwarz adaptive wavelet method for elliptic boundary value problems"],"prefix":"10.1090","volume":"78","author":[{"given":"Rob","family":"Stevenson","sequence":"first","affiliation":[]},{"given":"Manuel","family":"Werner","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,11,13]]},"reference":[{"key":"1","unstructured":"[Bar05] A. Barinka. Fast Evaluation Tools for Adaptive Wavelet Schemes. Ph.D. thesis, RTWH Aachen, March 2005."},{"issue":"233","key":"2","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1090\/S0025-5718-00-01252-7","article-title":"Adaptive wavelet methods for elliptic operator equations: convergence rates","volume":"70","author":"Cohen, Albert","year":"2001","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1007\/s102080010027","article-title":"Adaptive wavelet methods. II. Beyond the elliptic case","volume":"2","author":"Cohen, A.","year":"2002","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"issue":"5","key":"4","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1002\/cpa.3160450502","article-title":"Biorthogonal bases of compactly supported wavelets","volume":"45","author":"Cohen, A.","year":"1992","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"key":"5","series-title":"Studies in Mathematics and its Applications","isbn-type":"print","volume-title":"Numerical analysis of wavelet methods","volume":"32","author":"Cohen, Albert","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0444511245"},{"issue":"6","key":"6","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/S0893-9659(99)00075-0","article-title":"Besov regularity for elliptic boundary value problems in polygonal domains","volume":"12","author":"Dahlke, S.","year":"1999","journal-title":"Appl. Math. Lett.","ISSN":"https:\/\/id.crossref.org\/issn\/0893-9659","issn-type":"print"},{"key":"7","unstructured":"[DFP{\\etalchar{+}}07] S. Dahlke, M. Fornasier, M. Primbs, T. Raasch, and M. Werner. Nonlinear and adaptive frame approximation schemes for elliptic PDEs: Theory and numerical experiments. Bericht Nr. 2007-7, Philipps-Universit\u00e4t Marburg, 2007."},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/s10444-005-7501-6","article-title":"Adaptive frame methods for elliptic operator equations","volume":"27","author":"Dahlke, Stephan","year":"2007","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1093\/imanum\/drl035","article-title":"Adaptive frame methods for elliptic operator equations: the steepest descent approach","volume":"27","author":"Dahlke, Stephan","year":"2007","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"3-4","key":"10","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/BF03322055","article-title":"Wavelets with complementary boundary conditions\u2014function spaces on the cube","volume":"34","author":"Dahmen, Wolfgang","year":"1998","journal-title":"Results Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0378-6218","issn-type":"print"},{"issue":"258","key":"11","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1090\/S0025-5718-06-01917-X","article-title":"An optimal adaptive wavelet method without coarsening of the iterands","volume":"76","author":"Gantumur, Tsogtgerel","year":"2007","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"12","series-title":"Monographs and Studies in Mathematics","isbn-type":"print","volume-title":"Elliptic problems in nonsmooth domains","volume":"24","author":"Grisvard, P.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0273086472"},{"key":"13","unstructured":"[Pri06] M. Primbs. Stabile biorthogonale Spline-Waveletbasen auf dem Intervall. Ph.D. thesis, Universit\u00e4t Duisburg, 2006."},{"key":"14","doi-asserted-by":"crossref","unstructured":"[Sch1890] H.A. Schwarz. Gesammelte Mathematische Abhandlungen, Vol. 2, 133\u2013143. Springer, Berlin, 1890.","DOI":"10.1007\/978-3-642-50665-9"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"1074","DOI":"10.1137\/S0036142902407988","article-title":"Adaptive solution of operator equations using wavelet frames","volume":"41","author":"Stevenson, Rob","year":"2003","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"16","doi-asserted-by":"crossref","unstructured":"[SW08] R.P. Stevenson and M. Werner. Computation of differential operators in aggregated wavelet frame coordinates. IMA J. Numer. Anal., 28(2):354\u2013381, 2008.","DOI":"10.1093\/imanum\/drm025"},{"issue":"4","key":"17","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1137\/1034116","article-title":"Iterative methods by space decomposition and subspace correction","volume":"34","author":"Xu, Jinchao","year":"1992","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02186-8\/S0025-5718-08-02186-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02186-8\/S0025-5718-08-02186-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:56:33Z","timestamp":1776786993000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-266\/S0025-5718-08-02186-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,11,13]]},"references-count":17,"journal-issue":{"issue":"266","published-print":{"date-parts":[[2009,4]]}},"alternative-id":["S0025-5718-08-02186-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-08-02186-8","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2008,11,13]]}}}