{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T03:20:47Z","timestamp":1785900047050,"version":"3.56.0"},"reference-count":47,"publisher":"American Mathematical Society (AMS)","issue":"333","license":[{"start":{"date-parts":[[2022,8,12]],"date-time":"2022-08-12T00:00:00Z","timestamp":1660262400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["1210391"],"award-info":[{"award-number":["1210391"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["11170050"],"award-info":[{"award-number":["11170050"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript negative s\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\n                                \u2212\n                                \n                              <\/mml:mo>\n                              <mml:mi>s<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^{-s}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s element-of left-parenthesis 0 comma 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s\\in (0,1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Proofs are given in the case of uniformly and locally refined meshes. Our findings can be applied to define local multilevel diagonal preconditioners that lead to bounded condition numbers (independent of the mesh-sizes and levels) and have optimal computational complexity. Furthermore, we discuss multilevel norms based on local (quasi-)projection operators that allow the efficient evaluation of negative order Sobolev norms. Numerical examples and a discussion on several extensions and applications conclude this article.\n                  <\/p>","DOI":"10.1090\/mcom\/3674","type":"journal-article","created":{"date-parts":[[2021,6,16]],"date-time":"2021-06-16T10:25:40Z","timestamp":1623839140000},"page":"183-218","source":"Crossref","is-referenced-by-count":12,"title":["Multilevel decompositions and norms for negative order Sobolev spaces"],"prefix":"10.1090","volume":"91","author":[{"given":"Thomas","family":"F\u00fchrer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2021,8,12]]},"reference":[{"issue":"6","key":"1","doi-asserted-by":"publisher","first-page":"1901","DOI":"10.1137\/S0036142997330809","article-title":"The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling","volume":"36","author":"Ainsworth, Mark","year":"1999","journal-title":"SIAM J. Numer. 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