{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:47:37Z","timestamp":1776725257817,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"234","license":[{"start":{"date-parts":[[2001,3,2]],"date-time":"2001-03-02T00:00:00Z","timestamp":983491200000},"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>\n                    In usual boundary elements methods, the mixed Dirichlet-Neumann problem in a plane polygonal domain leads to difficulties because of the transition of spaces in which the problem is well posed. We build collocation methods based on a mixed single and double layer potential. This indirect method is constructed in such a way that strong ellipticity is obtained in high order spaces of Sobolev type. The boundary values of this potential define a bijective boundary operator if a modified capacity adapted to the problem is not\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1\">\n                        <mml:semantics>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This condition is analogous to the one met in the use of the single layer potential, and is not a problem in practical computations. The collocation methods use smoothest splines and known singular functions generated by the corners. If splines of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 m minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2m-1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are used, we get quasi-optimal estimates in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript m\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mi>m<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -norm. The order of convergence is optimal in the sense that it is fixed by the approximation properties of the first missed singular function.\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01209-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:17:46Z","timestamp":1027707466000},"page":"607-636","source":"Crossref","is-referenced-by-count":1,"title":["Optimal order collocation for the mixed boundary value problem on polygons"],"prefix":"10.1090","volume":"70","author":[{"given":"Pascal","family":"Laubin","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,3,2]]},"reference":[{"key":"1","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3024-3","volume-title":"Complex variables","volume":"125","author":"Berenstein, Carlos A.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387973494"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1137\/0519043","article-title":"Boundary integral operators on Lipschitz domains: elementary results","volume":"19","author":"Costabel, Martin","year":"1988","journal-title":"SIAM J. 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Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"6","series-title":"INSTN: Collection Enseignement. [INSTN: Teaching Collection]","isbn-type":"print","volume-title":"Analyse math\\'{e}matique et calcul num\\'{e}rique pour les sciences et les techniques. Vol. 4","author":"Dautray, Robert","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/2225812985"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s002110050107","article-title":"An optimal order collocation method for first kind boundary integral equations on polygons","volume":"70","author":"Elschner, J.","year":"1995","journal-title":"Numer. 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Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"key":"10","series-title":"Monographs and Studies in Mathematics","isbn-type":"print","volume-title":"Elliptic problems in nonsmooth domains","volume":"24","author":"Grisvard, P.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0273086472"},{"key":"11","first-page":"623","article-title":"Annihilator ideals and representation iteration for abstract rings","volume":"5","author":"Everett, C. J., Jr.","year":"1939","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"367","DOI":"10.2307\/2006988","article-title":"The Dirichlet problem in nonsmooth domains","volume":"113","author":"Jerison, David S.","year":"1981","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1090\/S0273-0979-1981-14884-9","article-title":"The Neumann problem on Lipschitz domains","volume":"4","author":"Jerison, David S.","year":"1981","journal-title":"Bull. Amer. Math. Soc. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0273-0979","issn-type":"print"},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s002110050333","article-title":"High order convergence for collocation of second kind boundary integral equations on polygons","volume":"79","author":"Laubin, Pascal","year":"1998","journal-title":"Numer. 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Japan (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0370-1239","issn-type":"print"},{"key":"17","series-title":"Mathematical Topics","isbn-type":"print","volume-title":"Pseudo-differential boundary value problems, conical singularities, and asymptotics","volume":"4","author":"Schulze, Bert-Wolfgang","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3055015975"},{"issue":"3","key":"18","doi-asserted-by":"publisher","first-page":"572","DOI":"10.1016\/0022-1236(84)90066-1","article-title":"Layer potentials and regularity for the Dirichlet problem for Laplace\u2019s equation in Lipschitz domains","volume":"59","author":"Verchota, Gregory","year":"1984","journal-title":"J. Funct. 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