{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T02:41:48Z","timestamp":1768531308399,"version":"3.49.0"},"reference-count":29,"publisher":"American Mathematical Society (AMS)","issue":"236","license":[{"start":{"date-parts":[[2001,7,21]],"date-time":"2001-07-21T00:00:00Z","timestamp":995673600000},"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 propose a new consistent, residual-based stabilization of the Stokes problem. The stabilizing term involves a pseudo-differential operator, defined via a wavelet expansion of the test pressures. This yields control on the full <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n  <mml:semantics>\n    <mml:msup>\n      <mml:mi>L<\/mml:mi>\n      <mml:mn>2<\/mml:mn>\n    <\/mml:msup>\n    <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-norm of the resulting approximate pressure independently of any discretization parameter. The method is particularly well suited for being applied within an adaptive discretization strategy. We detail the realization of the stabilizing term through biorthogonal spline wavelets, and we provide some numerical results.<\/p>","DOI":"10.1090\/s0025-5718-00-01263-1","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:13:53Z","timestamp":1027721633000},"page":"1397-1416","source":"Crossref","is-referenced-by-count":5,"title":["Stabilized wavelet approximations of the Stokes problem"],"prefix":"10.1090","volume":"70","author":[{"given":"Claudio","family":"Canuto","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Roland","family":"Masson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2000,7,21]]},"reference":[{"key":"1","first-page":"1","article-title":"Wavelets on closed subsets of the real line","author":"Andersson, L.","year":"1994"},{"issue":"6","key":"2","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1016\/S0893-9659(98)00114-1","article-title":"Stabilization by multiscale decomposition","volume":"11","author":"Bertoluzza, S.","year":"1998","journal-title":"Appl. Math. Lett.","ISSN":"https:\/\/id.crossref.org\/issn\/0893-9659","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"[BCT] S. Bertoluzza, C. Canuto, and A. Tabacco, Negative norm stabilization of convection-diffusion problems, Appl. Math. Lett., 13 (2000), pp. 121\u2013127.","DOI":"10.1016\/S0893-9659(99)00221-9"},{"key":"4","doi-asserted-by":"crossref","unstructured":"[BCU] S. Bertoluzza, C. Canuto, and K. Urban, On the adaptive computation of integrals of wavelets, Preprint n. 99-5, Dipartimento di Matematica, Politecnico di Torino; Appl. Numer. Math. 34 (2000), pp. 13\u201338.","DOI":"10.1016\/S0168-9274(99)00028-8"},{"issue":"1-2","key":"5","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/0377-0427(96)00022-2","article-title":"Least-squares methods for Stokes equations based on a discrete minus one inner product","volume":"74","author":"Bramble, James H.","year":"1996","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"6","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and hybrid finite element methods","volume":"15","author":"Brezzi, Franco","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387975829"},{"issue":"3-4","key":"7","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1016\/S0045-7825(96)01221-2","article-title":"\ud835\udc4f=\u222b\ud835\udc54","volume":"145","author":"Brezzi, F.","year":"1997","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"[CTU] C. Canuto, A. Tabacco, and K. Urban, \u201cThe wavelet element method. Part I: construction and analysis\u201d, Appl. Comput. Harmon. Anal. 6 (1999), 1-52.","DOI":"10.1006\/acha.1997.0242"},{"key":"9","unstructured":"[Co] A. Cohen, \u201cWavelet Methods in Numerical Analysis\u201d, to appear in Handbook of Numerical Analysis (P.G. Ciarlet and J.L. Lions, eds.), Elsevier North Holland, Amsterdam."},{"issue":"5","key":"10","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"},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1006\/acha.1993.1005","article-title":"Wavelets on the interval and fast wavelet transforms","volume":"1","author":"Cohen, Albert","year":"1993","journal-title":"Appl. Comput. Harmon. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-5203","issn-type":"print"},{"key":"12","doi-asserted-by":"crossref","unstructured":"[CM1] A. Cohen, R. Masson, Wavelet Methods for Second-Order Elliptic Problems, Preconditioning, and Adaptivity, SIAM J. of Sci. Comp. 21 (1999), pp. 1006\u20131026.","DOI":"10.1137\/S1064827597330613"},{"key":"13","unstructured":"[CM2] A. Cohen, R. Masson, Wavelet adaptive methods for 2nd order elliptic problems. Domain decomposition and boundary conditions, Preprint No 98007, Laboratoire d\u2019Analyse Numerique, Universit\u00e9 Pierre et Marie Curie, 1998. Accepted in Numerishe Math."},{"issue":"4","key":"14","first-page":"341","article-title":"Stability of multiscale transformations","volume":"2","author":"Dahmen, Wolfgang","year":"1996","journal-title":"J. Fourier Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1069-5869","issn-type":"print"},{"issue":"5-6","key":"15","doi-asserted-by":"publisher","first-page":"1015","DOI":"10.1016\/S0016-0032(97)00033-1","article-title":"Nonlinear computation","volume":"334","author":"Seydel, R\u00fcdiger","year":"1997","journal-title":"J. Franklin Inst. B","ISSN":"https:\/\/id.crossref.org\/issn\/0016-0032","issn-type":"print"},{"issue":"3","key":"16","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1007\/BF01385864","article-title":"Multilevel preconditioning","volume":"63","author":"Dahmen, Wolfgang","year":"1992","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"3","key":"17","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/BF02238515","article-title":"A wavelet Galerkin method for the Stokes equations","volume":"56","author":"Dahmen, W.","year":"1996","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1006\/acha.1998.0247","article-title":"Biorthogonal spline wavelets on the interval\u2014stability and moment conditions","volume":"6","author":"Dahmen, Wolfgang","year":"1999","journal-title":"Appl. Comput. Harmon. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1063-5203","issn-type":"print"},{"issue":"2","key":"19","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1137\/0730024","article-title":"Using the refinement equation for evaluating integrals of wavelets","volume":"30","author":"Dahmen, Wolfgang","year":"1993","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3-4","key":"20","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/BF02072014","article-title":"Wavelet approximation methods for pseudodifferential equations. II. Matrix compression and fast solution","volume":"1","author":"Dahmen, W.","year":"1993","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"228","key":"21","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.1090\/S0025-5718-99-01092-3","article-title":"Composite wavelet bases for operator equations","volume":"68","author":"Dahmen, Wolfgang","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"22","doi-asserted-by":"crossref","unstructured":"[GT] S. Grivet-Talocia, A. Tabacco, Biorthogonal wavelets on the interval with optimal support properties, Math. Models Meth. Appl. Sci., 10 (2000), pp. 441\u2013462.","DOI":"10.1142\/S0218202500000252"},{"issue":"3","key":"23","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1016\/0045-7825(86)90152-0","article-title":"A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems","volume":"58","author":"Hughes, Thomas J. R.","year":"1986","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"4","key":"24","doi-asserted-by":"publisher","first-page":"965","DOI":"10.1137\/0729059","article-title":"Wavelet methods for fast resolution of elliptic problems","volume":"29","author":"Jaffard, St\u00e9phane","year":"1992","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"25","doi-asserted-by":"publisher","first-page":"221","DOI":"10.4171\/RMI\/123","article-title":"Analyses multi-r\u00e9solutions non orthogonales, commutation entre projecteurs et d\u00e9rivation et ondelettes vecteurs \u00e0 divergence nulle","volume":"8","author":"Lemarie-Rieusset, Pierre Gilles","year":"1992","journal-title":"Rev. Mat. Iberoamericana","ISSN":"https:\/\/id.crossref.org\/issn\/0213-2230","issn-type":"print"},{"issue":"6","key":"26","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1142\/S0218202596000328","article-title":"Biorthogonal spline wavelets on the interval for the resolution of boundary problems","volume":"6","author":"Masson, Roland","year":"1996","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"key":"27","unstructured":"[Ma2] R. Masson, Wavelet discretizations of the Stokes problem in velocity-pressure variables, Preprint Laboratoire d\u2019Analyse Numerique, Universit\u00e9 Pierre et Marie Curie, 1998."},{"key":"28","doi-asserted-by":"publisher","first-page":"606","DOI":"10.1090\/S0002-9904-1939-07048-1","article-title":"Topological proofs of uniqueness theorems in the theory of differential and integral equations","volume":"45","author":"Rothe, Erich","year":"1939","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"},{"issue":"1-2","key":"29","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/BF02123473","article-title":"On divergence-free wavelets","volume":"4","author":"Urban, Karsten","year":"1995","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-00-01263-1\/S0025-5718-00-01263-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-00-01263-1\/S0025-5718-00-01263-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:36:45Z","timestamp":1627605405000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-00-01263-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,7,21]]},"references-count":29,"journal-issue":{"issue":"236","published-print":{"date-parts":[[2001,10]]}},"alternative-id":["S0025-5718-00-01263-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-01263-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,7,21]]}}}