{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,29]],"date-time":"2026-03-29T21:44:39Z","timestamp":1774820679795,"version":"3.50.1"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"290","license":[{"start":{"date-parts":[[2015,3,11]],"date-time":"2015-03-11T00:00:00Z","timestamp":1426032000000},"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>We construct projections from <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H normal upper Lamda Superscript k Baseline left-parenthesis normal upper Omega right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>H<\/mml:mi>\n      <mml:msup>\n        <mml:mi mathvariant=\"normal\">\u039b<\/mml:mi>\n        <mml:mi>k<\/mml:mi>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">H \\Lambda ^k(\\Omega )<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, the space of differential <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n  <mml:semantics>\n    <mml:mi>k<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> forms on <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega\">\n  <mml:semantics>\n    <mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">\\Omega<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> which belong to <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared left-parenthesis normal upper Omega right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>L<\/mml:mi>\n        <mml:mn>2<\/mml:mn>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">L^2(\\Omega )<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> and whose exterior derivative also belongs to <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared left-parenthesis normal upper Omega right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>L<\/mml:mi>\n        <mml:mn>2<\/mml:mn>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">L^2(\\Omega )<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, to finite dimensional subspaces of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H normal upper Lamda Superscript k Baseline left-parenthesis normal upper Omega right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>H<\/mml:mi>\n      <mml:msup>\n        <mml:mi mathvariant=\"normal\">\u039b<\/mml:mi>\n        <mml:mi>k<\/mml:mi>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">H \\Lambda ^k(\\Omega )<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> consisting of piecewise polynomial differential forms defined on a simplicial mesh of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega\">\n  <mml:semantics>\n    <mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">\\Omega<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. Thus, their definition requires less smoothness than assumed for the definition of the canonical interpolants based on the degrees of freedom. Moreover, these projections have the properties that they commute with the exterior derivative and are bounded in the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H normal upper Lamda Superscript k Baseline left-parenthesis normal upper Omega right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>H<\/mml:mi>\n      <mml:msup>\n        <mml:mi mathvariant=\"normal\">\u039b<\/mml:mi>\n        <mml:mi>k<\/mml:mi>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">H \\Lambda ^k(\\Omega )<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> norm independent of the mesh size <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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>. Unlike some other recent work in this direction, the projections are also locally defined in the sense that they are defined by local operators on overlapping macroelements, in the spirit of the Cl\u00e9ment interpolant. A double complex structure is introduced as a key tool to carry out the construction.<\/p>","DOI":"10.1090\/s0025-5718-2014-02827-5","type":"journal-article","created":{"date-parts":[[2014,3,11]],"date-time":"2014-03-11T17:02:31Z","timestamp":1394557351000},"page":"2631-2656","source":"Crossref","is-referenced-by-count":39,"title":["Local bounded cochain projections"],"prefix":"10.1090","volume":"83","author":[{"given":"Richard","family":"Falk","sequence":"first","affiliation":[]},{"given":"Ragnar","family":"Winther","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2014,3,11]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492906210018","article-title":"Finite element exterior calculus, homological techniques, and applications","volume":"15","author":"Arnold, Douglas N.","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/0521868157","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"21-26","key":"2","doi-asserted-by":"publisher","first-page":"1660","DOI":"10.1016\/j.cma.2008.12.017","article-title":"Geometric decompositions and local bases for spaces of finite element differential forms","volume":"198","author":"Arnold, Douglas N.","year":"2009","journal-title":"Comput. 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