{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T00:25:29Z","timestamp":1777508729710,"version":"3.51.4"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"240","license":[{"start":{"date-parts":[[2002,6,14]],"date-time":"2002-06-14T00:00:00Z","timestamp":1024012800000},"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>An essential part of any boundary value problem is the domain on which the problem is defined. The domain is often given by scanning or another digital image technique with limited resolution. This leads to significant uncertainty in the domain definition. The paper focuses on the impact of the uncertainty in the domain on the Neumann boundary value problem (NBVP). It studies a scalar NBVP defined on a sequence of domains. The sequence is supposed to converge in the set sense to a limit domain. Then the respective sequence of NBVP solutions is examined. First, it is shown that the classical variational formulation is not suitable for this type of problem as even a simple NBVP on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary. A new definition of the NBVP is introduced to avoid this difficulty by means of reformulated natural boundary conditions. Then the convergence of solutions of the NBVP is demonstrated. The uniqueness of the limit solution, however, depends on the stability property of the limit domain. Finally, estimates of the difference between two NBVP solutions on two different but close domains are given.<\/p>","DOI":"10.1090\/s0025-5718-01-01359-x","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T14:12:48Z","timestamp":1032531168000},"page":"1339-1370","source":"Crossref","is-referenced-by-count":31,"title":["Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions"],"prefix":"10.1090","volume":"71","author":[{"given":"Ivo","family":"Babu\u0161ka","sequence":"first","affiliation":[]},{"given":"Jan","family":"Chleboun","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,6,14]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"key":"2","doi-asserted-by":"crossref","unstructured":"[B1] I. Babu\u0161ka, Stability of domains with respect to basic problems in the theory of partial differential equations, mainly the theory of elasticity, I, Czechoslovak Math. J. 11 (86) (1961), 76\u2013105 (Russian).","DOI":"10.21136\/CMJ.1961.100444"},{"key":"3","doi-asserted-by":"crossref","unstructured":"[B2] I. Babu\u0161ka, Stability of domains with respect to basic problems in the theory of partial differential equations, mainly the theory of elasticity, II, Czechoslovak Math. J. 11 (86) (1961), 165\u2013203 (Russian).","DOI":"10.21136\/CMJ.1961.100453"},{"key":"4","doi-asserted-by":"crossref","unstructured":"[BCL] C. Bajaj, E. Coyle, K.Lin, Arbitrary topology shape reconstruction from planar cross sections, Graph. Models and Image Process. 58 (1996), 524\u2013543.","DOI":"10.1006\/gmip.1996.0044"},{"key":"5","unstructured":"[BPHN] C. Bajaj, V. Pascucci, R. Holt, A. Netravali, Active countouring of images with physical A-splines, TICAM Rep No 99-03, University of Texas at Austin (1999)."},{"key":"6","first-page":"285","article-title":"Sur les inverses des \u00e9l\u00e9ments d\u00e9rivables dans un anneau abstrait","volume":"209","author":"Hebroni, P.","year":"1939","journal-title":"C. R. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/0001-4036","issn-type":"print"},{"key":"7","isbn-type":"print","volume-title":"Finite element approximation for optimal shape, material and topology design","author":"Haslinger, J.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0471958506","edition":"2"},{"key":"8","series-title":"Mathematics in Science and Engineering","isbn-type":"print","volume-title":"Design sensitivity analysis of structural systems","volume":"177","author":"Haug, Edward J.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0123329205"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1112\/blms\/13.2.167","article-title":"Approximation by smooth functions in Sobolev spaces, a counterexample","volume":"13","author":"Kolsrud, Torbj\u00f6rn","year":"1981","journal-title":"Bull. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"key":"10","series-title":"Springer Series in Soviet Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-09922-3","volume-title":"Sobolev spaces","author":"Maz\u2019ja, Vladimir G.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/3540135898"},{"key":"11","doi-asserted-by":"crossref","unstructured":"[N1] J. Ne\u010das, Sur les domaines du type \ud835\udd11, Czech. Math. J. 12 (1962), 274\u2013287 (Russian).","DOI":"10.21136\/CMJ.1962.100515"},{"key":"12","unstructured":"[N2] J. Ne\u010das, Les m\u00e9thodes directes en th\u00e9orie des \u00e9quations elliptiques, Masson\/Academia, Paris\/ Prague, 1967."},{"key":"13","series-title":"Studies in Applied Mechanics","isbn-type":"print","volume-title":"Mathematical theory of elastic and elasto-plastic bodies: an introduction","volume":"3","author":"Ne\u010das, Jind\u0159ich","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0444997547"},{"key":"14","unstructured":"[R] P. J. Rouche, Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Albuquerke N.M., USA, 1998."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01359-X\/S0025-5718-01-01359-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01359-X\/S0025-5718-01-01359-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:03:55Z","timestamp":1776726235000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01359-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,6,14]]},"references-count":14,"journal-issue":{"issue":"240","published-print":{"date-parts":[[2002,10]]}},"alternative-id":["S0025-5718-01-01359-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01359-x","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":[[2001,6,14]]}}}