{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:10:29Z","timestamp":1776838229223,"version":"3.51.2"},"reference-count":40,"publisher":"American Mathematical Society (AMS)","issue":"349","license":[{"start":{"date-parts":[[2024,11,17]],"date-time":"2024-11-17T00:00:00Z","timestamp":1731801600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2324364 009098"],"award-info":[{"award-number":["DMS-2324364 009098"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2201487"],"award-info":[{"award-number":["DMS-2201487"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2324364 009098"],"award-info":[{"award-number":["DMS-2324364 009098"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2201487"],"award-info":[{"award-number":["DMS-2201487"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    A new overlapping Additive Schwarz preconditioner is developed for high order finite element approximation of planar linear elastic problems on triangular meshes. The new preconditioner results in a condition number that is\n                    <italic>\n                      bounded independently of the degree\n                      <inline-formula content-type=\"math\/mathml\">\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                          <mml:semantics>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                          <\/mml:semantics>\n                        <\/mml:math>\n                      <\/inline-formula>\n                      , the mesh-size\n                      <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>\n                      and the ratio\n                      <inline-formula content-type=\"math\/mathml\">\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"lamda slash mu\">\n                          <mml:semantics>\n                            <mml:mrow>\n                              <mml:mi>\n                                \u03bb\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\/<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mi>\n                                \u03bc\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:annotation encoding=\"application\/x-tex\">\\lambda \/\\mu<\/mml:annotation>\n                          <\/mml:semantics>\n                        <\/mml:math>\n                      <\/inline-formula>\n                    <\/italic>\n                    . The resulting condition number is reduced to roughly\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"6.0\">\n                        <mml:semantics>\n                          <mml:mn>6.0<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">6.0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for all values of the parameters and discretization parameters on standard test problems. Crucially, the overall cost of the new preconditioner is comparable to the cost of applying standard domain decomposition based preconditioners.\n                  <\/p>","DOI":"10.1090\/mcom\/3926","type":"journal-article","created":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T10:23:31Z","timestamp":1699007011000},"page":"2067-2102","source":"Crossref","is-referenced-by-count":1,"title":["Uniform preconditioners for high order finite element approximations of planar linear elasticity"],"prefix":"10.1090","volume":"93","author":[{"given":"Mark","family":"Ainsworth","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Charles","family":"Parker","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2023,11,17]]},"reference":[{"issue":"6","key":"1","doi-asserted-by":"publisher","first-page":"1395","DOI":"10.1137\/S1064827594272578","article-title":"A hierarchical domain decomposition preconditioner for \u210e-\ud835\udc5d finite element approximation on locally refined meshes","volume":"17","author":"Ainsworth, Mark","year":"1996","journal-title":"SIAM J. 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