{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T14:35:05Z","timestamp":1776868505546,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"230","license":[{"start":{"date-parts":[[2000,5,19]],"date-time":"2000-05-19T00:00:00Z","timestamp":958694400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n This paper provides a framework for developing computationally efficient multilevel preconditioners and representations for Sobolev norms. Specifically, given a Hilbert space\n \n \n \n V<\/mml:mi>\n V<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and a nested sequence of subspaces\n \n \n \n \n \n V<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n \u2282\n \n <\/mml:mo>\n \n V<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n \u2282\n \n <\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n \n \u2282\n \n <\/mml:mo>\n V<\/mml:mi>\n <\/mml:mrow>\n V_1 \\subset V_2 \\subset \\ldots \\subset V<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we construct operators which are spectrally equivalent to those of the form\n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n =<\/mml:mo>\n \n \n \u2211\n \n <\/mml:mo>\n k<\/mml:mi>\n <\/mml:munder>\n \n \n \u03bc\n \n <\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n \n Q<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \n \u2212\n \n <\/mml:mo>\n \n Q<\/mml:mi>\n \n k<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathcal {A}= \\sum _k \\mu _k (Q_k-Q_{k-1})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Here\n \n \n \n \n \n \u03bc\n \n <\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \\mu _k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n k<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n <\/mml:mrow>\n k=1,2,\\ldots<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , are positive numbers and\n \n \n \n \n Q<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n Q_k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the orthogonal projector onto\n \n \n \n \n V<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n V_k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n \n Q<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n Q_0=0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We first present abstract results which show when\n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {A}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is spectrally equivalent to a similarly constructed operator\n \n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \\widetilde {\\mathcal {A}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n defined in terms of an approximation\n \n \n \n \n \n \n Q<\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n k<\/mml:mi>\n <\/mml:msub>\n \\widetilde Q_k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n \n Q<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n Q_k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , for\n \n \n \n \n k<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n <\/mml:mrow>\n k=1,2, \\ldots<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We show that these results lead to efficient preconditioners for discretizations of differential and pseudo-differential operators of positive and negative order. These results extend to sums of operators. For example, singularly perturbed problems such as\n \n \n \n \n I<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n \n \u03f5\n \n <\/mml:mi>\n \n \u0394\n \n <\/mml:mi>\n <\/mml:mrow>\n I-\\epsilon \\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n can be preconditioned uniformly independently of the parameter\n \n \n \n \n \u03f5\n \n <\/mml:mi>\n \\epsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We also show how to precondition an operator which results from Tikhonov regularization of a problem with noisy data. Finally, we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition.\n <\/p>","DOI":"10.1090\/s0025-5718-99-01106-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:17:31Z","timestamp":1027707451000},"page":"463-480","source":"Crossref","is-referenced-by-count":52,"title":["Computational scales of Sobolev norms with application to preconditioning"],"prefix":"10.1090","volume":"69","author":[{"given":"James","family":"Bramble","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joseph","family":"Pasciak","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Panayot","family":"Vassilevski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,5,19]]},"reference":[{"issue":"202","key":"1","doi-asserted-by":"publisher","first-page":"447","DOI":"10.2307\/2153097","article-title":"New estimates for multilevel algorithms including the \ud835\udc49-cycle","volume":"60","author":"Bramble, James H.","year":"1993","journal-title":"Math. 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