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Comp."],"abstract":"<p>\n                    The fractional differential equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript beta Baseline u equals f\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mi>\n                                \u03b2\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L^\\beta u = f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    posed on a compact metric graph is considered, where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"beta greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\beta &gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L equals kappa squared minus nabla left-parenthesis a nabla right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03ba\n                                \n                              <\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u2207\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mi mathvariant=\"normal\">\n                              \u2207\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L = \\kappa ^2 - \\nabla (a\\nabla )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"kappa comma a\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03ba\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\kappa ,a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript negative beta\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\n                                \u2212\n                                \n                              <\/mml:mo>\n                              <mml:mi>\n                                \u03b2\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^{-\\beta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L 2 left-parenthesis normal upper Gamma times normal upper Gamma right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0393\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0393\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L_2(\\Gamma \\times \\Gamma )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L equals kappa squared minus normal upper Delta comma kappa greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03ba\n                                \n                              <\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03ba\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{L = \\kappa ^2 - \\Delta , \\kappa &gt;0}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are performed to illustrate the results.\n                  <\/p>","DOI":"10.1090\/mcom\/3929","type":"journal-article","created":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T14:46:34Z","timestamp":1701096394000},"page":"2439-2472","source":"Crossref","is-referenced-by-count":16,"title":["Regularity and numerical approximation of fractional elliptic differential equations on compact metric graphs"],"prefix":"10.1090","volume":"93","author":[{"given":"David","family":"Bolin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mih\u00e1ly","family":"Kov\u00e1cs","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vivek","family":"Kumar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexandre","family":"Simas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2023,12,27]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"1541","DOI":"10.1103\/physrevb.27.1541","article-title":"Superconductivity of networks. 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