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Comp."],"abstract":"<p>\n                    We propose and analyze a robust Bramble-Pasciak-Xu (BPX) preconditioner for the integral fractional Laplacian of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s element-of left-parenthesis 0 comma 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s\\in (0,1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    on bounded Lipschitz domains. Compared with the standard BPX preconditioner, an additional scaling factor\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1 minus gamma overTilde Superscript s\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mover>\n                                  <mml:mi>\n                                    \u03b3\n                                    \n                                  <\/mml:mi>\n                                  <mml:mo>\n                                    ~\n                                    \n                                  <\/mml:mo>\n                                <\/mml:mover>\n                              <\/mml:mrow>\n                              <mml:mi>s<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">1-\\widetilde {\\gamma }^s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , for some fixed\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"gamma overTilde element-of left-parenthesis 0 comma 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mover>\n                                <mml:mi>\n                                  \u03b3\n                                  \n                                <\/mml:mi>\n                                <mml:mo>\n                                  ~\n                                  \n                                <\/mml:mo>\n                              <\/mml:mover>\n                            <\/mml:mrow>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\widetilde {\\gamma } \\in (0,1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , is incorporated to the coarse levels. For either quasi-uniform grids or graded bisection grids, we show that the condition numbers of the resulting systems remain uniformly bounded with respect to both the number of levels and the fractional power.\n                  <\/p>","DOI":"10.1090\/mcom\/3857","type":"journal-article","created":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T10:32:31Z","timestamp":1686306751000},"page":"2439-2473","source":"Crossref","is-referenced-by-count":2,"title":["Robust BPX preconditioner for fractional Laplacians on bounded Lipschitz domains"],"prefix":"10.1090","volume":"92","author":[{"given":"Juan","family":"Borthagaray","sequence":"first","affiliation":[]},{"given":"Ricardo","family":"Nochetto","sequence":"additional","affiliation":[]},{"given":"Shuonan","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Jinchao","family":"Xu","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,6,9]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"472","DOI":"10.1137\/15M1033952","article-title":"A fractional Laplace equation: regularity of solutions and finite element approximations","volume":"55","author":"Acosta, Gabriel","year":"2017","journal-title":"SIAM J. 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